Content-Type: multipart/mixed; boundary="-------------0306061723140" This is a multi-part message in MIME format. ---------------0306061723140 Content-Type: text/plain; name="03-270.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-270.comments" For later versions see the web pages of the authors. ---------------0306061723140 Content-Type: text/plain; name="03-270.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-270.keywords" tiling dynamical system, Penrose tiling, pinwheel tiling, substitution tiling ---------------0306061723140 Content-Type: application/postscript; name="text14.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="text14.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: text14.dvi %%Pages: 16 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentFonts: CMBX10 CMR10 CMSY10 MSBM10 CMMI7 CMMI10 CMSY7 CMR7 %%+ MSBM7 CMMI5 CMSY5 CMTI10 Helvetica CMR5 CMEX10 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips text14.dvi -o text14.ps %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2003.06.06:1715 %%BeginProcSet: texc.pro %! 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All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 22 /floorleftbigg put dup 23 /floorrightbigg put dup 80 /summationtext put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueXX 5000774 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6d5649664203ab24acb938d58d246707ffe7 d62f04bec4b70c21ef75beb2b812622b3c74e969d72d3cd11bd7106294a99caf 0b1629bc7d4de6b96ca82930831d64575f23f4ad06a0e45e315b1d392411be8d 6d73c998789ff258a07a3c8c2057325784514c845500bfd1a971310cfc11d41c 1a167dbd5ff012c60add4e87325f6e5299032a839de65fb1473a166aae1876a4 414a434f22c1d241591fb36f857df6fa930608750ffc0c54f44994662b1f00f1 400bf752ea8d83ffc4cb77a290bc2d99981ae59a191748ba5c7ba1a9d2583fd2 1398452b6ff5d83a059f7eadcd2ef744e9dd22bdf9c79d049bf06835e878c32b 7765c69bdd8ef4deb4ea7cfff4cf9354a4ddffa689de961d16772491c7afbd7f ffde42400764c68e954ee5c455a5687959829bc3b319b2147deaab3628662c80 30c5e02fea09609abe4eaa12e217bc3af673f1bc36a7039eb13fcacb4218fe0f c5a3f9452d4edf46cc91db67b624d4f2d37502fb9f11af4da18ca40b61097f95 d44329375467ed13c5cb585ec53f62b83ef9502cc755af44bf32b87b8ae9f3f2 f8dbf72dab90acafbacd280db6aaffaefdff6d5eff26669bac56280a950560e3 d01714ba38659b0e4e28bf8cdc1f15ff7e3ea9d95a6afbbe3d8e6905aa6d531d 6b6dcab89e2036cc492c65f76ddcc3cfcff33fde3d9ffd10f7a90530fd5c7683 4329baa902e701ca42717476298c9ff511b1941244bb1dfd0e5cab92a43970cd 1bdb292eadd055c97972844fdcdb0aa06401e7b2632b9d134f96bce0064860f3 9859da7a9a82be0a840672a1331212752ae1b521b693b384a72dba8d37a86eaf a0341e7fc5f6ef7a8c566aacb91e8bf5005c18837894c4eacff41dba54e071ca 83f80e7f256d4fe2fde6e2408ec819f8fecbb87409ec86c0fd05e839b3b52ec3 3684bde34965630ee053f43af0fb729761d3b7e24c396fdc52fcca3e52536af8 2a4f2baf35642c71f01cbfa89b9ec6b71d76b7239d4a423b1f86a8b02f46c0ef da564c7ab4fffc50ca4ea0f86cc7f9a3cf2c5909360329c2551cfcb37a969a8c 2db87ecc509d1714232378898a72e4f4e2c8048b6b5a0c7427fa4a65d71fce18 1734dd4ffe64a18b0908fdcf9e7b100e2b3962d395cc1d7deef53835f19dbe1a a59381c1553a057cfe1c324e581f7a56451b60593d651c74fdf08aca10eba04f cabb89b7dacf4389efa29e6a48747c9e8cb53e372378614a9ca497f1fdfc255b e4a6e77f4047974b23d379009b2951666e77fbf9170beb9b9050071cbb7bf048 0f2f4b02626101d4caf0437549da4b33a3f8a1684057bf2b5ab6fb3db297f145 c2f1c7667b0692b02db9a9bb9d465af66c67e2f9cbccb3eef34e00ecdf299f77 24d436327313af5f823b8029e251923badfc1276230ffd0820ec61b26ab47a28 999a8ca0efd40e5d05d634ea4fe7e13d84f7b1fdd4e95bb6a5dfd332c13649db 18c876d09035107abc6dfd3460914c50ea6c5870bd4c063e9af2c9cf6992060a 4ba12ce643eee492 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR5 %!PS-AdobeFont-1.1: CMR5 1.00B %%CreationDate: 1992 Feb 19 19:55:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put readonly def /FontBBox{-341 -250 1304 965}readonly def /UniqueXX 5000788 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 97e17cb027f5c73fdbb56b0a7c25fc3512b55fe8f3acfbffcc7f4a382d8299cc 8fd37d3cea49dabdca92847af0560b404ef71134b0f3d99934fc9d0b4e602011 b9cfb856c23f958f3c5a2fbe0ef8587d1f5774879c324e51fcb22888b74f2415 50d7401eb990d4f3a7af635198422283cac1b6cd446ddbcbd915db9bff88844e 784c6bf7389803d9450b0c21756a017306457c7e62c1d269f306bd3402e266de fc3b5e7d8a8d2f5bf0fe6ddd40d07391df4fad4a6018dce29a2b8f692b29f202 3a7c0e66de8ed85c14f1f8492167357f51a7e84cc5d92e0fee4a71813d2ffb26 445026f89b4787516ecd1afc78f8bd19e91e9ccc9402e8c36d2449c1ff850a8a f61165aac3fe931332dd28e261b91b05edd18f44ea7d58a8f35fe88493b64aad 6bfac3a0136215fc2f4ca8e91d70c5010e6f4013e6d63b44f6fb00afdbd7cdb4 5ec5b1d9736f45cefc8a0124b815987cebd81bbe0d44dedb2d5ef37923b3d551 abb6a1cefca7868fc7bf3814ef7d7b6b1ae6e869cb77aa29e3d90b12b0dc3ff6 ec945922b5899bbf2f12e92731486d2ef1230c528bf8d7e0ae09ad7632a38966 5963de49d1ef3d65bc483e4a577b927c940f5e121169ba52f6576c85793e5fd7 7f5863c488e55bdefd5d8b2514795533aabbc12e7f51816c7e1484f1c441aa9a 66a5dc77158e79cd6692d299b95e8058b35e771a6972d2b5eb1cd6d2bb8e835d 361a6eac0c90c906aea1cd75c4412a1339dc1439f93e59917311bf20af4f2bc8 bba5a4626f75f28a47d4595c37f4019df480d9385975159142618e22e964f019 d88125210e2ef21d65baf500adf802297b24eebbf4e93c617a613f856bd0d45f e17a9480980ef83950bdb8ebf208c895ebc4a7132f0765b7c19a332ebbf8a3c9 ab65063a42e19ca9d43dec3de7aba46b9a5d24fb93d615c0a1726ab7e31c378e ab77150858535b91362ab0e2b968dee1c539964126d937f9d4980ccefa92bc81 b054996a65a13995c9c9d819ff05511cfc89cbfe67447c940405001d61be633b b771eff383fc6977ec06af30f981504260de7b9ec97359974734f495ac073935 fb30e3700679092ef5d971b6b2da0e15960f2391fa6b63c33e7f56b4a9b86951 1d3686a5e8e613a69a0add8b8c73b127ddcd397a48d785079e786d24b6868c73 c8868545d5bdf432f8f31894c97a60150969cb73e1e76a399c601763359f5a5a 7c6a5aa7a71c76fc13cb9cfde253d76c906b30c308c9e7a72d64757de4a91ea5 4e44f4ce198306ec50cefd94ad932e3e7646ee275251f9383fe2536f444d8d0d 980a6a25721548831ef18a4dfe478c6589918a46ebb64fe7727fe6be9a5c5f0b e5e47cb3a55adb8cb10b56d8f56af3d40d672c347be7e61688869455d572569f 6bf6b96d66d63f0be5076d479eb90e776c6eae00891c4c499c6e9a6506d8ca62 90ce7d0c9f440f25d2d5a6a2a874ec429a6574416dfe36a7a320a6f9e2124351 9d39bb4bb418afee48d12e0874ccdae2fcf36925f8bb6a97dd0a59c862c51b85 3a712b0f740e1cb50e877ac84f954660c4c3ce573ddef3724041b787db50b040 0f6fee3d1c4a46f2a2fe6a601cf2f6632e8bd03eb1e5bece9f1a382e149d8c3f cc9afd9e2ad7e8f167d2aca0190863c3236233cd3f23e1323783b3396c5bcc14 63366ee6a64bb1b0421353fdc433887454f887e1fe15844bb711e4624c7255a0 77242f1936defe19d03b84b58c65eee1eff08f186dd4677848c50aa411a65035 e5aa4623a80ef2b855ea578873763cdccee3e604c71a26316af45cefd560a29d 1de749f62dd048fe136a6a0dd56ae8626fb27110670ecd5c3dc98a4c1c7db63c 82266961646e079fed3df55f0296ae458efd876cb16a20a3d1794d57e208f693 50df30eb8bd8dd5d0e3f9e575bfe3c814ec9b276518da9c36bc98a8643bbc094 80e4b3562e21215b646daab2ad71aedc403d23fde2bf421d4829a631d62d4c11 0d3da1e0fd205833b55e870ab7e9e21ac322488029541d4fe01a07be20d12341 9e70d799f3e791c7571623ff5694a1e2a5f39709fbc182353ae713b1a5f26a16 0ae50d0a69a4447a193c8eb6bf13e70e5175d59b6ed1dae92818326cd40477f9 1a9ca170c697691efc2ddfa39ce2d8780c93bdb4d524f7ab0281b3f3b01e419a bbeac96b3d5547d201fd31c91ed3526712ef5c792859da271923912ca32dd23d 674a4306ab90da86b1100aa8904903c85f1c578856068834c221e2af39b4c3c7 4a92a70e005446ac17992e58124192cd2d36e475b1d4ff7191999e1bab998d50 f68cbcfc44cf7c004e2a71cd1b29bf920214dc0d0634061d6ee0eac39471d163 d21e732729c4f11009f1e8c9e8009d58ca2c5c2672d25826780eda9444f60a2d af654515dae339a14a3e6ee07156b67129384faf9864bcd822f79ddcbeaa31d4 2f39cc072bce449d404bfff05d57ff02a0e033582334e1c3290f210b 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI10 %!PS-AdobeFont-1.1: CMTI10 1.00B %%CreationDate: 1992 Feb 19 19:56:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 45 /hyphen put dup 46 /period put dup 65 /A put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 73 /I put dup 77 /M put dup 83 /S put dup 84 /T put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 120 /x put dup 121 /y put readonly def /FontBBox{-163 -250 1146 969}readonly def /UniqueXX 5000828 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3ffd20326947b1065649b533eb5e9e3 a88a87fe987918a687950b26f647d6e08bf14d983ed1d4a8ca1e2da8d985f944 2eb916f5b6ed0697ac7c33e1e36a3bf460d34ce45f1631871097cb04f18e3889 4cf4ac1538eb19481311d24fe3be7beaa4a3730e8b4831fe59d6d9ce2e46116b 629c7ba2f9ce3ecba2f43bc162a5a077ca1b2882a42afdcec3f4b75b5d63e0bc 8e5dc95257766d8ea467ad9cbaf47be60f797580cced6884b3a68f70c91f4fdf 80fd00ed9139e7f480dc3a76af72ad9b434187730bdfaefe4cbfe5c7edcaaf24 9204fd703011932e5a3c27be468b7dab69daa18dbbb6335ab8ddfdc607961c7c 02582763fa069d43563ba17704029945cf42fcb19cf78c51df0ec4d851086d43 c1ab38e865ef36865fff3a08b01cbcb070ba4893f7482dc7819ad03d337e520e 7d8cd83bb8ac7aff4df36751f4f12aaaea2b6d7260c09a26e987c904003e0723 7dbcc4a8e4e85cd3259f40d3caf55c742cbb40ad0db1044c20a4f122a63f7b81 1d945bfa69662189ce4d089464d2ead9f6ed60b581b04114d5d45e97be27a576 35154e9b8464f2a0df18da855eb51b8f64d3054b0b2e84621f025a0776d16fed b988a43d2d64c32768dd106b9f541ae8e68ac0bb7399e16f4d4ba074b2152d0a 4e9b5bfc21bfa0d531e28165e518ffc054dd4ed0d74eca079d20e79670419f92 e03bcbfeecf3554d661596f6f29e12170804d15a250d917ffe7a8b6e8fa53047 1ed97ba56481589a65ca13619129f1e26affe6c22f0c38540f8ce37efb6e9e9b 60b51a5a9cb4454fd7c7f92f2f9437159687ab590f35b946a54491a7739c9269 e0f1fb48b129811c82edb3171cbd5105fe65906974db8a7da92f4f43dc7f6b36 6981146d86d23613a0b38d9d6191db9fea0172e982cb1cc50e382f69da1b377c 1be9407f88a2a761b56f398467a057f35d18c01cc377e006182c891ec80272e2 2e3a5c7123cd27eccdc87392a9fdf63bb40f02b7649cfc53ef933c6518744cb8 9cbbd437830ef5b1d58b00b67bbcf4426dc1e6ef583b0b4ad3aebd741e3ff0f7 29ae271f032019b1e7268043fd215be590f9cf24da14a90f1683afcba8006cef 5d5632c2481fbc1bd8224e8952debb7c4596a237e5505ac143180956ca1f603e ec70cdb4a33e1c4ca1744adab17c56e805c5a53b130f561cf0b9bfc0a5c7fabf 7f588769860fcf171d1ec8dde503be1d5bc14661ee0b58b3e0b44105ffdcdde7 226998c87327263eb3068901eab9346948123542562a692428d4c5bdf5376b91 a13f35f17be1b94f7c3199e94a6c3e989e681c2588ed3e657dcea2443b82f907 68652b8145e74cc472473b9cb7cfabd4e53023af0527612139049372cad19773 0b170718395b66e66638dff0a6d76e2613836a85a98ea3ae1d6c955061d98040 e12a74905f97daddc356346a1f9921f49af1b4579d417825cd6063d6e6bf2d06 df6f7bf2959ff80fd8f68d5682400e31dbe42c9c99b84fa95453554a236aa5a2 367c9d8d8a3b0ad5b1c92c6d42bc7cbbb8db1c631b9fabf5513fd811f64b9603 e47d017dbd1c7dcc06aa52d941ccb93ed9cfbe24daf29b6114cc801b1e0932f0 73e4c2b8988ce8be9dc8f18853a45b970f9d5bf4caaecc49fc17e830e463f504 03d58764c5aa78048b13cff9b724ee728f3f74c98f649625f4c1993ad66c720a def198e4ada5655468a6005876c5191a3f76b4e002538682be6cc4516b7aa137 54b8ee21be5d91120c493ca157489b5b497e28119d2af83e7f1c57c592de8ad5 a93f944e03f51cc00d1b4f7bf419af4bcac1835b7bdcbd10fda756bee60c34b0 e52368a21badfca14f3ac5093e4b65a3cfde5993636e76f2f84b1632715864d7 0ece4cf1133b764b3728d64950d7a7ba6643eaa9c8c539918d7eaaae8fb85019 8ab10953919794a72cc062bc160ed7ef0715476165d07e3be1c5c50b6c12d21e ca64e9c2322ee499e4ff02cd61dc24d58899b0e1db5bc92ad972772952bd2198 737685e7e06089ed6ba253c526663b77e881f426012423673b5547d85750745d fcd4f46d423db63c625399b5c4823758354e378e1a3aa715cc0692e8ecf2fa83 2acbce987dc358ec0f765cf20c4217444f66170664e77cd8a19337936d1d25f0 513c949249f093fb413707faeecb648f816ac589b44b2cc576084dd9ddf69ddb 153e48bd40724168a0bebeb7d2d8d1aba8d3b99fcaef6cf1ae25656da92e6ec6 57d3dbdddebb52596786130a630c4ef4f0c528541141982bf2ffb4d7a4419ea4 efca201e5a92a90119531b3eb9b91483843e90b2b4864e3df65c56ec3c71c8bf d5ea81b0c0e62eed8fc1b53d5889642a7af37c704b2cd78921c3f1ae48379bc9 3495f51753372aa9093e91140ab369c0872fb73a353f767ae5ceecff0676579f b5740d7f96238ceae40e8b2b225df7f8768eab10bd3589ed6387b8fd5646f9f8 ddf488a95b989b749095db0752ce6cb9269f74d867673043f5c530d0a5725ac7 671d708fc73981c245457c18342142a3d192f1798910a7490df3efd4fc355850 d561e0ca7e91375c780f11a77997e571574848cecd841b61c323dca84a12d1f8 9fcf7ba89b2a84d1ff374c559958d05bd95f6639ebdf935c08d7edc0b9c63d0a 483c64bcb5b38a2686e4c3d2bf2328c1d224fab7c500591f7528d6a25fea4585 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%!PS-AdobeFont-1.1: CMSY5 1.0 %%CreationDate: 1991 Aug 15 07:21:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 48 /prime put readonly def /FontBBox{21 -944 1448 791}readonly def /UniqueXX 5000815 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f059084178f5ab59e11b6784decd2fb36dec8f 7478da853762e00bc9c52ba86ef50d222ff142abc18e4fb95b86b3e57d9390ca 86e8de6347bf7f48f93322a372c5d7830521b14e8fa0771961b027c81365426d d95d16b950e216a90f28ce806244ff28dd969cb512a23e05045e7b2bcaf7d6f0 ca8b51494f2ec0681bdfde60420628d9bce98a0ca1ecca9140f452a88db19a30 ed46ae881b098c843a51d06f9a153489357fd24de6a0c36cb5a9eb9b3b64dd75 18f83018ae7cefa1c12fd205df6880a705e28457af7b869a6dd28f6b37d82cc3 e51cba348dcdb34efb119b8ce5a996f15492b0f5d34538850a2c1b6a509fe5fa 015eceaf28b6f9385526e98445876548d0f2a09cfea14b84be3721b51afa2185 90c7b9ca643c7f4fe67cd8985f27855c40ae7ea592fadfbba29301071039d18a 8a6880cc0bff22fc8af50fd2802aff5a96381fb4bef240dd26b34a5461188ca0 c85e4b3f803f94dce14e8ddb2284e22abfe148560aff784967ed795332cc0995 80b762f864b2c15bc640a4cd7f6927c7dd95aef777fd226254b8e6e1b8ec6483 2750d9929627c4cecf9f5c8feb57a2c7ee5d1315593f3fa16c7c7292a594ce14 2c396d9b787665569dbd0f710822ae3155a3aa53d6f055cb19e897b235b65bce 4d507a4d41edc6a09dff1f737f2711478e6e998218b7f3e7c1aaf3938f68e9cf 9ea46e4a2904c459281cc156f5ce80d6356e357011cd0bc383318129bc8ce38a 758c05d1205192a457f4b1b31df97baec7611905cac0 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI5 %!PS-AdobeFont-1.1: CMMI5 1.100 %%CreationDate: 1996 Aug 02 08:21:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 67 /C put dup 84 /T put dup 100 /d put dup 105 /i put dup 106 /j put dup 107 /k put dup 110 /n put readonly def /FontBBox{37 -250 1349 750}readonly def /UniqueXX 5087380 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311e252234d51741e18db3a68e8ad10242 29e5817a10e796a78d2c7f7c1f50961b9a57aa604c9f821dbf5a9295197bc666 31678d7d2c7e1f8f2151ce0c29efce46270570f4301c5dad1b38884732e53dad 05dfc36b020e726ca575f71f04abf2b49e1f1d6d2a08a477658ca9a48f1c8cce 14382f0a42201de56e2821ee2b0e91a818b1b753d810ec2428dae5153b66af66 9442192024becf743a8f6264a58a0b3aa823510a4957d1bab3e1b83135362888 3e392f49926471a8021fdb48637baf0abcf31204b1b214f48c99f44af7782323 69d772ae4f49be53a1bf584747bf29b94f530d59a5be5370b8dd0691e32a21ba 9d0d592cf3bb50a99a3ecd18d6c4114cb4cfecc49705da90d7edb10e3a28f9eb a49347a09d31eec896f2a28e9607d99d42914aea16545c16e136bdd0b75fe486 61e24c7a9e9b233a76754c760822d0a88c575e08ae3e3f081d7bfb80ec4c03ac a08b65fb6474f71fc95c7c2462f2eb3754e647c29498288f2ced0bd3a2bca5c3 2f80a61fa667802d376cacd34de1f9572d1b3ab09d6f352750db4a768f869638 a87457889aa502850427b40e1294c5ba33e6b17532b2d13b5f50e71582f93daa 231e86597872a2a968127afb0f36257a76ce1eb24e6c4f29fce2aaf6532fb5cf df84da03f03a45b915209f67d398c7c4f7a3b0623750259087a478dca684b29e a8b7f52060bae9394369ed1cecc645d113cca3772a8dc36c6d916b5c5e65454c 0a188a33395fd3da67ad0a31ddab6a8434dd80f0bd0ea1a2a62eecc331d88b3e 369552daa1c575a98fe84383548626b65be7c24ae7eb61159f149872cfd87cf7 6c1a7c155c7186c7600e58cb61b8cb0c5dea98268a39e928bf6fc53985eaa763 2b28c1eb3d82e4d3fe225223f4266d2b6147e344aad2d4884a9308aadc7801af 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26aa26944143d9a8ffdb1876a25ba4bcf7bd5f441bf5ba64c955f86d9ea6535a cdb03a1aa6b9082c7eb7aeaff95caf5d0e071f6f4867730d154f98fcd93d4b83 4e032fc5dda278459fcd4fa171fccbd464e686c1553620446482ae6cbdc6f22c b3596294400eb513829f77136a53df12533b344a1eff86c0b341fa5446d5a08d 12409415958f8dd68d6f6eb67ce0493cec9c4cc76de3cdbc5de47fb7d68596a3 f235f40db44b1e1eec24aa73f0d6bb1ef5b161eed2a3e0dad44ecfb3535b78b4 6f41cb838862df2d8f784d9140c631a751489fd1492f705a167733a158f998db ab22bb3434f517b01f53064b3e17454e542527df0fb53ca4ffbd3a2784959df4 2caecf9cce5192c9c4158e7d2d9345c9fee7f7fee51df048089a639a8cc77e17 5f3fef7f4ee11d46c96133656ca3d83f4961faf457d15df86218cfae41e338be 5b31287e61e92c7bf7aedcfce005cd84b5348c61b38a675de34c689f098d75e3 375e76e9a6ec4b1ac428cf84c6ffec1af0806119e073edb9f42ccba4a6ad0dc8 9998129459c2e03b5a281e2dbce760f783ef268e1eba8b815a4bd7f5dc95170d 143a94ba6cf4ea8ef60f7ff3affe8eeeca3d0d14b6479b27c7b4a7c238f0c7eb c223703e532ab048e8d9aa8e56af300a6d428614114390f98bdcc815df90f165 70c4d6dc22a68462cc2d67efc57f582a50a59c2603d6097decb0429047b6d91d 504fe9100209c83075997633851007728d8d651a4920633a77dde4b541591b6e b4ce43e4bf52c1d08bb6dd52897203fdeae191d8a6475d3d6d38074e8e06c561 6cf4e865bdd4e5603a26e92701aa867880b7bdbd869eec0f5b35485d2f24721e f58ca8bc7c946d7903ca62da13939ad69cffaf111196a594770dca1a5da860b4 d3d425bfb4dfb9e5fa094671703145de5fa8dba08b1129286c396d9e46fcf01c 57b69767b8391536719f4df7c3c04ce0c2fb95dd0328d2da445b8a710acf6f13 f42c7c40d9929403b2c7654f21c69604cbae52594f71e116e97a60d70d6b2605 2ecde4f5b2da76de1ccbae2a8ed8801bddcc92ddb37ae4f504702fc5ff921e4b 81adc58a81a0744c1829f97d3cd16d9e530aaba093cf52a996d9c0b6f16656b3 0da8574bd5c67a7755a1c1d2d334217909a835a00c597b6ef9a3f9d5875fe5f1 3488815cae6fa49ee52cac5f8e71209f109d6f5ab635a1855684244e9ec431b8 4bbb2db84252b39c816d6df2adc946c7dacef3c318c5424b0e78c4dc32eb6029 a9ea095104508d510fe3819a6ef26a83748fa1803f6b04f1494f30a65109b66a 67bd2ee1332a49e4868c17a0d0afa111058a722bf3c0143b4eaf4f37789cb9dd 9d73fd5b8ea3f31ed486215877b12dfd1e7680db7cb422b499745ec20a56374d 1b0ca69ab2a82194adaade1c05eeb0597b8bf940aa6f17be63298075ff8ce502 4f4249bc2d60fbe0baba4f63fd9833e4d7e543a729ac3a9df475b91e21c8b205 a3304ba5862e64893203f4275bebe012bda32f3e6eeb6290dae79e30dc6c580d 8772357c571232ebaed83003484bf51b179c5e6663067e27488c1964e6c6abb7 96b64cd4ef0993e717d5b49c07b4880705a85fa57245a4dceb0e9b5dbbc31b1a a6cdcee18ac5fe68ba0e8691c77407d4f6e959ec88ba72154d1a94932acdc5b3 a00c906795abf2b26388c1e7088e94efdc8cc4ae357daf446c2552a8c8eadf29 f2b83e28b14c2984737be8f9c427a53f6fc1674eb927b121c548a833329d0702 54e9246cbf68a04d5967aad30ba30e1f212bf87118bb4d3e351aacba426da4ee 592eaf0120175ef24fdcee27ec738a1110721f0b6dfd2f4fc883f313fb8b7f9e 52ae1f6dfd99d573c7f5f990c8e56892659f2da5f1934010370d3e703fd5c7be 929ae7ccca08be78541460e54d0811e81c499d5d9d8384784bd13fcae68685e7 a11ad54eb9b26f9869f5aa90e09371a2041d5a433fae7db69dbf1911323eafa5 681bcbf794053a81e1be8b9202c2716f1352d9d7fdb811876f270c6b4f715ecf 97aaf4e70648bbbe3d78641ec2df65292f980b16e107e1915d04036db9bf792b 38ff3c96384f09d6beaae0310a56cbab67368163c7dc77711b07de6181f48c96 54 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSBM7 %!PS-AdobeFont-1.1: MSBM7 2.1 %%CreationDate: 1992 Oct 17 08:30:50 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM7) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 90 /Z put readonly def /FontBBox{0 -504 2615 1004}readonly def /UniqueXX 5032014 def currentdict end currentfile eexec b5e2457af4a6caaa4536453c498a618ecdce7b5f6f2d5944265309f833bd3d50 3df8bd14c071f404d2177500681a801856cb8c9c9cf0112c9bd17ccba9ab8944 1759a320b2078df0fafdd3e357994bd81740d1a7522ddfafb75d58f5def046f7 084d10fa2f702e570b09a7073d70efe37b48c355a7089772238b7b4538c4912f cf6703a14fa416057ea2e1b94455913dd271b42846ee10d78270a8118e7e4bcb 8deb048bb280c240d59409d47046c890e5f79e5dd4b0be65a28a77aeb53dd33e ac506a6da87bb55e329f433627e06c3c94b078838b908538d93f888e504e25e0 3a8ccf7d2c02122ab3d485525564295390d1154e15fee35edff4eb016fadd32c 26d1b4c09de87943780954cebc4965aa890fe1f80b03eb07103e13d468245c7b c26455f229e8859585c6eeafd9474acdb27290fd62950dc52f893c99a2a614d4 2a7e3618266f3fb0c68a01a30373e048385ffe7dd5844e208a55d1a9263ab2db cb9b19038724f95af00349840157be45cb337871b55a304b87ade64b10ff0eba 21aa0eaeef74aa01f049303a8b25de0acc26bb3a33fcf267f479a2a73f7dece3 2aa23083687322b417fe86e8f5afc2136d045ad6b392364811a17630a94ff097 038eedeedb8da4ebf8fa5a96ce27a2ebf771d14f4a11fb1f9f48cdd207209934 885ef9118463552984b0fd3cf15d55de967fbb051549dd65f60dfe07f299690e 58252f032bca78935b7d2df852ed688bd13dd171b91cf7bd46d5433aef9f50f1 41a2517c4912b837793f84ddcf1116f70f7e9b1e6ec752dbc9183d4791d92602 397bafe29ac6f879ceb72c673c8a1aa0135d6f92d0db7791f9a2f1a5b55347e0 3e63723efd258754f5f4a9ad35873a98d54b6c54b17a39892dc1d38660597448 fdba9258493a0db0d560c01d49f2d889a99eaa8a5b5919cbd33f6b3d97bf27b3 e0b94b48d5d21d6c29aee4faca71bd775bb9defa37ff627f1e4926f10d59313e 77d81a511027ef9a8ed5b7fa9b5bd67758cd13bab6ac18411c974a65d0c90085 67bf5860d0930d7f43e557ee45e76ecb2806d21f4732bb282ac8b591f2b8775d 4b3ce2a37ebb371cb74a3cc513bdd70a0115de4760c21f034f5080d769f7ff83 554af524184702a02e534d1eb98dd3f7c5d51900409354dea37de4ac7cd42348 7c653dd50f6cda46266680119b22d86d17c4b63f13c00e0eac9bc346f92854e8 c2970254f55a231e056af8a18a32de3acaacfc722ce4dda7d5fd74cecc69b5d0 1dce2e5a86bd847bbdfebb0e9007c718727faa6637472e3b8629a9cd03a313ee 22c14b7894f13bd06dd32ba0d808628fa71320fe44a67b8792d1aeef6f8a07d9 f2420591a424b7f4afb559be14cdce92756b9b577787165b7e457d01873cbc72 4e6697a9ae81d712dcbc1638222177007037dbef6bf5ff1d976722a633a07ad2 3e9cf134f23b18fe315e877f941e35dd33870c3ed3d843e6e8673ae59215a0ae ea9a3c9db8b6a5df631b65b8481e1eabd0e9035dad89c59fee35158a6d5c411e 2f39658f385a4bc895af26e9b4d5173a36fcfea72083fafc712f213916e32c1d 946e26e88591b761f6cd59afd0a3974206f46c90207d83e18fd75f51be61ba8f 8ae7a83afd69a767d1 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR7 %!PS-AdobeFont-1.1: CMR7 1.0 %%CreationDate: 1991 Aug 20 16:39:21 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 61 /equal put dup 79 /O put dup 82 /R put dup 91 /bracketleft put dup 93 /bracketright put readonly def /FontBBox{-27 -250 1122 750}readonly def /UniqueXX 5000790 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 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0d2349162ae3ac93144eee1e665a1289105318fdfe86b6e76251cb25adc967d3 d0b97fe5e279e1161736ab22b4ca510b964342383a840defd38f96a7280e6ac1 34e48d740607ff2e7804164a16d47735864db8463a486ce4314ce68a4abbfd0c 1d8090a1146e51c1147146f1189098b0ac1b31819e481ca297840873961ddd87 af46c4d818cc252607802320dde80926c63221841035992073004f7dc1ec8b82 581e2e62ed4d0cc3ef0599b2dee151270a188a7649e96e60b4578236e03bb332 6e18e077fa8af24e434ae05a3139f24b6da32f4ae82a86570b3f933c7c8ab503 7809a99ab190d89e32c8f9a62383c0b3ef7ec5822b4938ad704a2ea9525789c0 44b896d7ab110480d052885813827cd1e58ea250608e1126edfb2b4f6b51fe21 85719a4aac3483334d83bcb22707b77b07325e674f5569592205d5e5dbaf3da5 ccdf8b72f115bbf7ace9f8f2572010494cd46b0314d8675f188d6e48716e54ef 67e5548d0f20b4aedccfbc0345b8144fb78072cd78c9fbe2070a82a085c7aa0a c2bc6706e2f3e7fd7f6b68e42e1f960d40af323826329c3acb324c32a7428ee7 e3da90b8e0016099dcbd1117bb9895ecfa433b814003cef2971578cdb54c9e9b 816bcc2b21c2fbf84526a980bfd4b804fd0058314eabaab6aebaa05e15899255 dda5bcd04749cd1885b65c575f0989208108db6221610e6472bbad37f5eebf3b 39bd1783fb4940aff9e2d3fa0040f7387f567c2d79f18d708e072736dc426433 b6b5013d4c3302e35c774cdc17b49756f649547baff8b8401c0e936596317d89 1aef6517e2af4d25faea5ed9f7029d7aed004bb25a92010613b50b37ceea02ad 0aa35d812239bc7f1a625339a1b577020640044b4b1846124ecb6cc19067cca9 5504b616d611c42c06b1ec695adaac26482da1fcfecf8eb1fb0794364bbee22b add019b9fe0857e4c052cc611495c3f5674f499e69d91f41866adab347a4734a 9fa68f665088a181c0fd080ade0c5b91cfdb0c15e31d06e62141bb3d70b81f01 bcb7edb426766ba6869d5f02a720da42ca6fcd1ba97a43f9a9c5af6f2357af34 1df90bbca67a26a2ef7ed93a131b3527dc21330eefe1164ddc6d294bd2f14207 395aa25df77ab2b421ebf23273ccf526bc93d9aae503e6afb2b8862c610e970a 7b35d32dda6c0dd5407611186bb31c233e408775c5840092811607166e0fb192 82b4693fb47e4ba29c937f042b6ad9e47e837e2d0690353625ebf4ea9103940b 1afb8b6da73c656d33d8c833a48114e5a1d5f433d2ea146aa12d3536724e6557 c891aa9c4dd420db65984e50f11df8f237e8cfbf79b7509e38dfb4eabfbfa002 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f652eeb008863255971af04848b0cb69a0c780904b2bf4cd55e4d5f65e415e8c 24f0c181329fc716c24eeecb58fe8f8761e58eea12845d0c078842f4dde290ae 1caf1deeb51d894e99ac8485133e9d828baf7941e1a2fd0b4ef7cf56580223c5 37e1f2c24d1dc65273741fd0b320432604df8f9bc51d46a944a88e35e4b1fa22 3e5369fad2054357f031171d27eca1c442c2259ed154e0dd1ebfe31f514f0fcb 8a5f3e576dabc549f4bdacfcf96587739fc6ca9ada9f59fd5bafe5add2946893 6fe76255c61cc3506d5aa267687bdf4f66ceacbc3a70f85b3a8d3e49746c60c3 0686575582d56aeeb7a062969804fa8afc55bb57a1dc83a11a0c70b8941407b8 a4334a422219681d8d3abb32213a42b623a6436ed2f9f2e1a554cd902c11f560 c0669e86246a980f704d086cbe0353b046764298d7e1fedfd4b8017da70daf39 d6eb018a7744ebb3dbfba2b79e852dd21287acc3788bd234780c95de9c36ba30 e7a3e6eac1673f72b75ef5f9a5d60371c584a0c309d51b944849d3398d7be5ea 14923f61d526c249818ffef68e0cf0bd3a95b387baa7f71559ba20be8c2bcb57 c4f7c39c4c1654c872cbffe43ffcf3533399c720403a2ac0e48ca1234a010c36 2555b299617fb6b392d679a428e7ebaf18ca5e78739ea678f6e5e295158cdc1e ecd41825332bddb1b4fa1414831e18cbf6730c3fc2d93e4ec3c95b8047374111 550b6f5b5485df10ea2711ae2e54c0c4a1ea7628f994ac57bcd7847ccfc7c092 929b7bda6cfe89601b0ec81013d1d58759454edbe41fe627c7d502a74892a636 3f0a78cce08b225b427932dc47779922eeaaf12e68ee676213ab12b6292dd7aa 9b4d601ca5e31e647555c02a2eb9817963de1b8f42c59696eb39312cca1bc71f 120cfea62ca8d1673550f54ae63cdd3758292976536d18770363d6cac02756 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY7 %!PS-AdobeFont-1.1: CMSY7 1.0 %%CreationDate: 1991 Aug 15 07:21:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 14 /openbullet put dup 20 /lessequal put dup 21 /greaterequal put dup 33 /arrowright put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 65 /A put readonly def /FontBBox{-15 -951 1252 782}readonly def /UniqueXX 5000817 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f73e9b697ddc0351763500e23c4d170ea8a2d2 75ad0b6b23097f02fa803c1f46f9ac10fe0e527c431b11b1b9a50e874896cca6 17fe3e255960846bc319ca68b9b930a45d5d0fc36b352c3443ab6f7793b38d2c 7fb9c409ab7980ead55e9f8f6b83c70088d1fa6d7f2214da8928e40b96334ad2 72d1b5db9d1355a579b26a8dac21f021e393d7d837a95deb8de92ed818a8b80b 98ee9cdd8a3c57ad7d6eb7a82f43eaa6ec520eefa7302e750edc4b495f70bba2 24a2182eec37f1a1c5dc8e20973007150237daa6bf03d498826777eeebe09014 ebfe05a8731e1a421cfc513d47d673e3d94d805f5d0668ffa265e0962cb5a75a 4a93d9b87124d146d8836482f73a24389afb8a74364f22077cc3fd6015cd4469 c34470c86925433a77fb3c838a1a7158139e96980de23ff461382df32c6cb4c8 59d292d187a7c919a1c3e51647cecd04d7e091b936aaf37b758d1de4f68467fe 5d2d6655526e99bbd552d02a34ca9c8f4727aa4fb653cad9f3da27e0f390ee8a f6f37f76d1165c106a31ab163c8a9c966a98a46cd04cb5be09f36f284fe51ed4 d0196eaa3de6aa21825b5d64cc56fdf0ca77c5cbc555930de9ad20123cb434aa 7cd2ee9e41f2c44de6d3ca71432c75b36c77fe477ac966cd14eeb7b5f9d881d2 3a29c1f73b0b1508b8735470142f74118987c001a8712d6b781284e7e0d9c6dd 7fb074e799ee9a44566e3851a4181635fd628c624f62350b2b700ca4254b105d 52e531d59b7617124b6897936a4ab22b71017757ffae312c7f464fc68f490314 75f700de9e6ae334515bd7e79e4c07bfce3979a245d85d1bb80c084ffe229000 5d9411cc00980201c746eaa7c1737a047d120ec1bdf92bdfb066462f78a6c4af 3e88bdb0ccb411e207b880ee8c0b8f841a353a61d1d168f9d289ce0e4473d851 b69961c5a7d152b9b703047f3a308d6dbaf91127b9a430e8e6f8832acf57a528 e1a01f279d2772ef7e7b5f6a9598bf59c64f33d6993b04ce4fa99ccacc92ff33 73d50569c44e15cbefbce80198243d8eb7a6f2dee26b6c1c7563d88a0a6b136c cdf4e484131d70dbd25378930bfe8d94a0d939d480df8a78b62c20e9d83cc6ec af68cd14fe124e6d610f8235dbe6ca56ee809f7bb773d22fce426ea620cc43a7 75b36152d637325d3ce1015f2ed58c43b90678d4ab4dd46c74c763f217480f48 9cdfc9957c94c959f0e19d2ce7ba734fa86e4dd3fbae0c32de03c5e01f187759 2fb7dd5cf12077daa657087433fd1d95bb444edc33ab828ed95a227a4c1b30b7 bbc2cb6b9181bbe8ab226d1bc87518d18b01330dc9e5f04cda8c6710170d4623 564a7b252823f73f80f3bc00b4bed6b9b544267b792bda4e85f08379848556d1 1e71c5fb85a1ce2bdbfac216f9927abd927bfff4744c6e2d43efc6e911d1646f 43fcd283c84109cd0956b44fb110fae897d44f790eb8dfca34b66f699685da8a e359d258010b5d0e8f8f297aab1510c41e937d98272005c24e83ab25c3afbb95 dbf81af53be00d79d769807fd125130fed032fadafd28710d71145b1003b241c 801c335905d13656354229bfcb122b0d08206144926c9f41946966d46b21ae9f 6a4457cd19e556030933b1d73fe639cfdc1852e2a5c983cfa46fe78a092640e9 ec5ac309a0371334cda4093c544a48e203308ab903f38151da910c278706d43e e359fd8e836ad5d298b5cbe778baacfdb5acf19e1c3a8472651e9c67909ee5fa 4a80dcffd3209dcf1c693a39b3eba1ee7546dc556278e1b717ecd3e88c56d698 2612066147f4cc731af8067fa7e344ad6c5424cce1f160e14aca40c544f75872 b37ab2fe29f090fdda4417065b1c3493e253b0ea7712a903bfa23f1b867ae032 01cd3b12905e9332257154aae0eb082aadf45d2460cb470a74985dd6e18d8adb 23f26e76e0941ad7c52334397ad2ce87675b47500f3625b53ecb75312fc621e6 1c6a0e6a7b454db14f0556013b6b3a5dd0a6dd2e10949c84f29129ca7aeebb97 63987ef933690949bcaadb376e9110826783c7cf4f01382a7216d3ece2315b88 85258f145b0474dd689f114711ce0e8ecd66b5e25cb0ba57e156bb2648d43c82 c77b4f30d391a566f47a197b2fa4936419d4a8a67b55d7887455ccacdbb77ea0 9d202a5478a5cc5b9b677baed73f382c49a29670945d55e0144395e854941468 6b6aff2422a4b311b0549f2f57e570ec2d9e63e7d3b49dcc3be1e25b065fcfaf 3be7ea3b724ee9f95728371d2aedde389fe233253c655164f55c9ca6245fbf41 90a07f791ebdc5c526f4e8db988b6e199f8066ad5a 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 15 /epsilon1 put dup 17 /eta put dup 21 /lambda put dup 22 /mu put dup 25 /pi put dup 26 /rho put dup 28 /tau put dup 30 /phi put dup 32 /psi put dup 34 /epsilon put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 64 /partialdiff put dup 66 /B put dup 67 /C put dup 68 /D put dup 70 /F put dup 72 /H put dup 73 /I put dup 75 /K put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 83 /S put dup 84 /T put dup 87 /W put dup 88 /X put dup 96 /lscript put dup 97 /a put dup 98 /b put dup 100 /d put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 109 /m put dup 110 /n put dup 112 /p put dup 113 /q put dup 114 /r put dup 116 /t put dup 119 /w put dup 120 /x put dup 121 /y put readonly def /FontBBox{-32 -250 1048 750}readonly def /UniqueXX 5087385 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930da4f8d58abefd45db119b10eb409dd89792 3c6e705479464a4b310b58348c4b42393988fef4925cf984423aaf65fea9f0e6 4629953bcf50b919d968d99bd185f83112b2759cc411764e9bde677f57c5ee5a c555448d5f81a16259ded1e11bf4119d53e8ab07a802df900d3a2d5ccc1c6876 d29c5e9effb7af3ef83400b0910c07873a8c56fa7b1eb1ba426043b00cc95dbe dc6e136cbbbcb3f67509964f9f281ebf81fe5b018122eaf66c4a838487e82e18 6e006093042848a903efb3469ab6b4049767aadb95c30408dfd58d8a10f4cb22 168decd9f3ee100f07b49aa44c92139b669cc312ba20192454eb2375be6284b0 26659d964b96ae82d4942e758027fcf23c25ed01115af27ce7f20efe2a822bb6 84004f20243a49c9e93301fc21b80815c033c3e2ba58ef53da2157d524b395f2 b37abca13bc6a2f42e824ab7e47106176b0d6db267fbb795ac7425582df2e3dc 55863468a9200742bd7b552c48f8cf58bc21343bd3b95abfa140f33f37c6f3f7 8b0d8a5154eb7c1f62ec598267f13e841a3e64172663935ac8b665d86540d316 ddece329c008049c5e74b27d59022c5515059bc3b89370b1bc6a169c888bb325 e0b74282d6f053a50da4024ed1e433271a32ab8c17d41c632b41cf9f3ecd5fe1 24daf7aea7ce8a63047b245822930d517df7baaafa69d2d17f7d93cebd45416d f3459365434123a76ccf883a4973bab19807e1f0c4bab03a45f7ed69ff2660ff 3963e4def11001eec4238c368a39d874fd30b3c14f4186ec7700fbce22abe468 eb32baf7b0164f8f21a9b5e706b91411faaa44467cb2180ab03de375c1ab93e9 76e11ab92fdd4a7280a6d1a8dc65c4c89d04c8ed6988045a2a3de9a4c7b1efd3 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI7 %!PS-AdobeFont-1.1: CMMI7 1.100 %%CreationDate: 1996 Jul 23 07:53:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 21 /lambda put dup 30 /phi put dup 32 /psi put dup 59 /comma put dup 61 /slash put dup 67 /C put dup 68 /D put dup 69 /E put dup 72 /H put dup 75 /K put dup 77 /M put dup 78 /N put dup 80 /P put dup 81 /Q put dup 83 /S put dup 84 /T put dup 96 /lscript put dup 100 /d put dup 101 /e put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 112 /p put dup 113 /q put dup 114 /r put dup 120 /x put dup 121 /y put readonly def /FontBBox{0 -250 1171 750}readonly def /UniqueXX 5087382 def currentdict end currentfile eexec 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSBM10 %!PS-AdobeFont-1.1: MSBM10 2.1 %%CreationDate: 1993 Sep 17 11:10:37 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 69 /E put dup 78 /N put dup 81 /Q put dup 82 /R put dup 90 /Z put readonly def /FontBBox{-55 -420 2343 920}readonly def /UniqueXX 5031982 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6d56491bdecdcafdc988206c5a457a19270b 37d0ab776e03eaa7eb568eeab6b5e79dec03b0dcbf923a2aa8e4f4deda2cb043 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9a3822a0015a071ac8e83345310a3c175c609332e78240d31eeae40999b599f3 c0fc72f5b785876f688fa7ab48b5aea896cc4b4976f78518aa7b4d671eb1b118 ec584da869b59ac1249788acce228c6794bd5fc932c26c3ae5f345e3637d5d95 7911f42b9a107550bda2a5ad5ba133220a11eecc7816eaed67807bdf1fa083b9 0f2cbb98e1014154b1d75f4b2f9189e303f312662c4cea907f205e44fada7953 fc54f1ef2f1b750dc674865fcf9f4b56687cec58e04f5ba27a5a678afc441420 4b37bc6d89bedfe76e6edf60a13fa0f7dd43642a25a548be7c82e797e7addec9 6641eb409da5a60db853cff6534d8456cefd1bd86b4c6bc8ca8b20ae0b12a310 3d 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 6 /plusminus put dup 8 /circleplus put dup 14 /openbullet put dup 17 /equivalence put dup 20 /lessequal put dup 21 /greaterequal put dup 26 /propersubset put dup 27 /propersuperset put dup 29 /greatermuch put dup 33 /arrowright put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 55 /mapsto put dup 59 /emptyset put dup 65 /A put dup 66 /B put dup 71 /G put dup 76 /L put dup 79 /O put dup 80 /P put dup 91 /union put dup 92 /intersection put dup 102 /braceleft put dup 103 /braceright put dup 104 /angbracketleft put dup 105 /angbracketright put dup 106 /bar put dup 107 /bardbl put dup 110 /backslash put dup 112 /radical put readonly def /FontBBox{-29 -960 1116 775}readonly def /UniqueXX 5000820 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f67052788e6703bb5ee49533870bca1f113ad8 3750d597b842d8d96c423ba1273ddd32f3a54a912a443fcd44f7c3a6fe3956b0 aa1e784aaec6fce08dae0c76da9d0a3eba57b98a6233d9e9f0c3f00fcc6b2c6a 9ba23af389e6dfff4efec3de05d6276c6be417703ce508377f25960ef4ed83b4 9b01b873f3a639ce00f356229b6477a081933fef3bb80e2b9dffa7f75567b1fa 4d739b772f8d674e567534c6c5bbf1cf615372be20b18472f7aa58be8c216dbd df81cc0a86b6d8318ca68fe22c8af13b54d7576fe4ca5a7af9005ea5cc4edb79 c0ab668e4fec4b7f5a9eb5f0e4c088cd818ecc4feb4b40ec8bd2981bf2336074 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4a00879da75bf9d82f1433173d5710cd37b7d5140e474bae930daa70321cef3a f1ab270809bbb761aee8fb3689ab7c5f7458fe79bd5ce0d305345550bbf8140d 3bdb5cbbec99ed28bc5e5531bcd67ba8ceba891d8fdd2a8222c95dfb4938d74a dc9bb43b81f0a7a193b23e6df861169547f993c92e21d37d1a6292138961e993 afdf3f787c39ef5892229f7d62d5a18a9f922efa2182dde7c54a32d6bd653150 64898efd256658ff2705f4ea69aa6da0880256c4d101665c4d2d9f60613e1cab 1ec1d8eaa26a1c7f7529d14e4e41e9e671edcda34ab79a421b5de8f11fd4931a f333335d8aed3486df0824e7494da20b0ae1dc35017a7e15f56385b6ad9a1010 773c4f109a94f6acc0a9a1254e8ff54c54eedad809e66be7e18bf9fb02b7a3ca 3daf2494b228d1585751045b46738c722317c4f015cdb6aed0b88151679f61 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 12 /fi put dup 14 /ffi put dup 19 /acute put dup 34 /quotedblright put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 124 /emdash put dup 126 /tilde put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueXX 5000793 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 97e17cb027f5c73fdbb56b0a7c25fc3512b55fe8f3acfbffcc7f4a382d8299cc 8fd37d3cea49dabdca92847af0560b404ef71134b0f3d99934fc9d0b4e602011 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60ca13863be33455b3f3c309c8d59a39d667c36acd84a938df28b89a5868cce1 812e57adc70534c4438a79e403a412983a8cc775c96db656d16b2c91026f40c6 96d6ef77e64b95511971f356885432458c5fa0f97691077d52821b67ed8ae411 ccc645cc25f87651cdd0798f3111e39a29f342fd1b81baaa2c59338c37d66efe 35aafd951d07aa0bbefff35fe564bef58801f6ca9cacbf795a249ed6923dbfd5 a93e4cc3e584cc5cdac1141b3df1a996dc7bf7fab08a3c6a97def23bcd29c179 7541320552e6b39390b7d1d08ca5f5e44f6fd7a4573b632c3f93da54973e9f4d 981d97aaa4ed2eeb230efb7fc6f5663c0e96255c977176f2c696ccda34933762 5cf4cf7962df1cae1a9c00b908539bfed54d2eeca6f23f0b1959c71b0ac2e996 87bdd0c86f39566b703dca7d27072119257dc71ab6b146d4df2cefb7139a251b a8b09db8d73937fc7d98f9d970f392277b2ddaf4bb527284f8f83c2b8c76ea07 afce404ad780d342cacd5afc398aad83516f66dc621999dcd43c89da8156ec4d 09f267a99d44735d3bc63591fa94fa75d9ed2096238b1da1f02d23172c658f8f c7b1f4cff504f01e06c0ff565ac7c159360343c56f7ecfff04543cc8e5174103 06725b89ee10a10996d0aae754cf69f67b725b9bedf514bf44de7b9ec973567c bd5ab41aef55b5fb00be130d417b68d1c91e32649d6bc8437b7c9874f272e397 3fe4e7f3c55dffd7f03be631e20a39ff5827e8f8cdb07e744f42a7c9 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX10 %!PS-AdobeFont-1.1: CMBX10 1.00B %%CreationDate: 1992 Feb 19 19:54:06 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 44 /comma put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 65 /A put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 73 /I put dup 74 /J put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 89 /Y put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 104 /h put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 121 /y put readonly def /FontBBox{-301 -250 1164 946}readonly def /UniqueXX 5000768 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 97e17cb027f5c73fdbb56b0a7c25fc3512b55fe8f3acfbffcc7f4a382d8299cc 8fd37d3cea49dabdca92847af0560b404ef71134b0f3d99934fc9d0b4e602011 b9cfb856c23f958f3c5a2fbe0ef8587d1f5774879c324e51fcb22888b74f2415 50d7401eb990d4f3a7af635198422283cac1b6cd446ddbcbd915db9bff88844e 784c6bf7389803d9450b0c21756a017306462c563d51ece66fcc9c831843ecae 1fefc1a232e2724f7baee428ae03aadb95c3035345c15e9922fe49e1f2cfc980 237316572dbc57064edac9b0db8913a5e2d45e97e19a91435ccf8adfc835b585 48e74b291446d689c7f2f8c4325e8356e974ae30c3e2977477baeaa33d141fb7 80f59351e84bfc88c87b3a4a1d25e168b9d134554f6a581378c7d2d6eca8ac09 045cc3a5e0ed86f147133f094029e3483bebc81cfde69942cf645ea20d0a2b64 5140ccc2f9cb4a3d969f4b0a79cd3329b440dfcff5285494406f3dd08d804c86 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Fk(Z)1291 4885 y Fj(d)1338 4921 y Fm(-action)g(is)h(the)g(natural)g(action)f(of)h Fk(Z)2832 4885 y Fj(d)2922 4921 y Fm(on)g(a)g(compact,)i Fk(Z)3670 4885 y Fj(d)3717 4921 y Fm(-)150 5041 y(in)m(v)-6 b(arian)m(t)42 b(subset)h Fi(X)49 b Fm(of)42 b Fl(B)1213 5005 y Ff(Z)1259 4975 y Fe(d)1307 5041 y Fm(,)i(for)e(some)g(nonempt)m (y)g(\014nite)h(set)f Fl(B)s Fm(.)70 b(If)42 b(w)m(e)h(asso)s(ciate)e (with)150 5160 y(eac)m(h)k(elemen)m(t)g(of)g Fl(B)j Fm(a)c(\\colored")g (unit)h(cub)s(e)g(in)f Fk(E)2211 5124 y Fj(d)2258 5160 y Fm(,)j(the)e(face-to-face)h(tilings)d(of)i Fk(E)3554 5124 y Fj(d)3645 5160 y Fm(b)m(y)150 5280 y(those)31 b(arra)m(ys)f(of)g(suc)m(h)i(cub)s(es)g(corresp)s(onding)f(to)f(the)h (subshift)g Fi(X)38 b Fm(giv)m(es)30 b(a)g(tiling)f(dynamical)1925 5519 y(1)p eop %%Page: 2 3 2 2 bop 150 100 a Fm(system)39 b(whic)m(h)h(is)f(basically)f(the)h (susp)s(ension)i(of)e(the)h(subshift)g Fi(X)47 b Fm(\(but)39 b(with)f(rotations)h(of)150 219 y(the)34 b(en)m(tire)f(tiling)f(also)h 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b(tiling)f(dynamical)i(systems)g (there)g(is)g(a)f(natural)150 1295 y(analogue)f(of)h(suc)m(h)g(a)f (represen)m(tation)i(whic)m(h)f(w)m(e)f(call)g(the)h(\\Little)e(Caesar) h(prop)s(ert)m(y".)150 1475 y Fn(De\014nition)38 b(2.)48 b Fm(A)34 b(top)s(ological)f(conjugacy)i Fi( )f Fm(:)29 b Fi(X)2146 1490 y Fh(A)2245 1475 y Fl(7!)g Fi(X)2457 1490 y Fh(A)2521 1470 y Fd(0)2587 1475 y Fm(b)s(et)m(w)m(een)36 b(tiling)d(systems)j(has)150 1594 y(the)g(\\Little)g(Caesar)g(prop)s (ert)m(y")g(if)g(for)g(ev)m(ery)g Fi(n)2015 1558 y Fh(0)2075 1594 y Fi(>)d Fm(0)i(there)i(is)f Fi(n)c(>)h Fm(0)j(suc)m(h)h(that)f (for)g(ev)m(ery)150 1714 y Fi(x;)d(y)e Fl(2)d Fi(X)525 1729 y Fh(A)627 1714 y Fm(suc)m(h)34 b(that)f Fi(B)1143 1729 y Fj(n)1197 1714 y Fm(\()p Fi(@)6 b(x)p Fm(\))27 b(=)h Fi(B)1599 1729 y Fj(n)1653 1714 y Fm(\()p Fi(@)6 b(y)t Fm(\))31 b(w)m(e)j(ha)m(v)m(e)f Fi(B)2326 1729 y Fj(n)2375 1709 y Fd(0)2407 1714 y Fm(\()p Fi(@)6 b( )t(x)p Fm(\))26 b(=)i Fi(B)2877 1729 y Fj(n)2926 1709 y Fd(0)2958 1714 y Fm(\()p Fi(@)6 b( )t(y)t Fm(\).)150 1893 y(Examples)45 b(ha)m(v)m(e)g(b)s(een)h(giv)m(en)f(recen)m(tly)g(\(see)h([RS2],)h ([P]\))d(of)h(top)s(ological)e(conjugacies)k(of)150 2013 y(tiling)27 b(dynamical)h(systems)h(not)f(satisfying)g(the)h(Little)e (Caesar)i(prop)s(ert)m(y)-8 b(.)42 b(One)29 b(of)g(our)f(main)150 2132 y(results)34 b(is:)150 2312 y Fn(Theorem)52 b(1.)81 b Fm(Within)46 b(the)g(sub)s(class)g(of)g(substitution)g(tiling)f (systems)h(with)f(in)m(v)m(ertible)150 2432 y(substitution,)33 b(ev)m(ery)g(top)s(ological)f(conjugacy)i(has)f(the)h(Little)f(Caesar)g (prop)s(ert)m(y)-8 b(.)349 2611 y(Before)52 b(de\014ning)h(the)f(sub)s (class)h(of)f(\\substitution")f(tiling)g(systems)h(in)g(general)g(w)m (e)150 2731 y(presen)m(t)35 b(some)e(relev)-6 b(an)m(t)34 b(examples.)349 2850 y(A)40 b(\\P)m(enrose")h(tiling)d(of)j(the)f (plane,)i(Figure)e(1,)h(can)f(b)s(e)h(made)f(as)g(follo)m(ws.)64 b(Consider)150 2970 y(the)42 b(4)f(\(colored\))g(tiles)g(of)h(Figure)f (2.)69 b(Divide)41 b(eac)m(h)h(tile)f(\(also)g(called)g(a)g(\\tile)g (of)h(lev)m(el)f(0"\))150 3089 y(in)m(to)36 b(2)g(or)g(3)g(pieces)h(as) g(in)f(Figure)g(2)g(and)h(rescale)g(b)m(y)f(a)g(linear)g(factor)g(of)h (the)f(golden)h(mean)150 3209 y Fi(\034)49 b Fm(=)38 b(\(1)26 b(+)576 3126 y Fl(p)p 659 3126 50 4 v 83 x Fm(5\))p Fi(=)p Fm(2)39 b(so)g(that)g(eac)m(h)i(piece)f(is)f(the)h(same)f(size)h (as)g(the)f(original.)61 b(This)40 b(yields)f(4)150 3328 y(collections)k(of)g(tiles)f(that)g(w)m(e)h(call)g(\\tiles)f(of)h(lev)m (el)f(1".)72 b(Sub)s(dividing)43 b(eac)m(h)h(of)f(these)g(tiles)150 3448 y(and)36 b(rescaling)f(giv)m(es)g(4)g(collections)f(of)i(tiles)e (that)h(w)m(e)h(call)e(tiles)h(of)g(lev)m(el)g(2.)49 b(Rep)s(eating)36 b(the)150 3568 y(pro)s(cess)27 b Fi(n)e Fm(times)h(giv)m(es)f(tiles)h(of)g(lev)m(el)f Fi(n)p Fm(.)41 b(A)25 b(P)m(enrose)i(tiling)e(is)g(a)h(tiling)e(of)i(the)g (plane)g(with)g(the)150 3687 y(prop)s(ert)m(y)35 b(that)f(ev)m(ery)g (\014nite)h(sub)s(collection)f(of)h(tiles)f(is)h(congruen)m(t)g(to)f(a) g(subset)i(of)f(a)f(tile)g(of)150 3807 y(some)c(lev)m(el.)43 b(A)30 b(P)m(enrose)h(tiling)e(has)i(only)e(4)h(t)m(yp)s(es)g(of)h (tiles,)f(eac)m(h)h(app)s(earing)f(in)g(10)g(di\013eren)m(t)150 3926 y(orien)m(tations.)41 b(\(F)-8 b(or)25 b(P)m(enrose)h(tilings,)g (and)g(in)f(fact)g(v)m(ery)g(generally)g(for)h(tilings)e(made)i(b)m(y)f (suc)m(h)150 4046 y(substitution)36 b(rules,)g(there)h(are)e(\\matc)m (hing)h(rules")g(whic)m(h)g(pro)m(vide)g(a)f(di\013eren)m(t)i(metho)s (d)f(of)150 4165 y(construction)e([G].\))349 4285 y(A)f(\\pin)m(wheel") g(tiling)f(of)h(the)h(plane,)f(Figure)g(3,)f(uses)i(t)m(w)m(o)f(basic)g (tiles:)44 b(a)32 b(1-2-)3376 4202 y Fl(p)p 3459 4202 V 83 x Fm(5)h(righ)m(t)150 4404 y(triangle)d(and)h(its)f(mirror)g (image,)g(as)h(sho)m(wn)g(in)g(Figure)f(4)g(with)h(their)f (substitution)h(rule)g([R].)150 4524 y(Notice)e(that)g(at)h(the)g(cen)m (ter)g(of)g(eac)m(h)h(tile)e(of)h(lev)m(el)f(1)h(there)g(is)f(a)h(tile) f(of)h(lev)m(el)f(0)h(similar)e(to)i(the)150 4643 y(lev)m(el)37 b(1)f(tile)h(but)g(rotated)g(b)m(y)g(an)g(angle)g Fi(\013)d Fm(=)h(tan)2084 4607 y Fh(\000)p Fg(1)2190 4643 y Fm(\(1)p Fi(=)p Fm(2\).)55 b(Th)m(us)38 b(the)f(cen)m(ter)h(tile)f(of)g(a)g (tile)150 4763 y(of)g(lev)m(el)f Fi(n)g Fm(is)g(rotated)g(b)m(y)h Fi(n\013)f Fm(relativ)m(e)g(to)f(the)i(tile.)53 b(Since)37 b Fi(\013)g Fm(is)f(an)g(irrational)f(m)m(ultiple)i(of)150 4883 y Fi(\031)t Fm(,)32 b(w)m(e)g(see,)h(using)g(the)g(fact)g(that)f (within)g(a)g(tile)g(of)g(lev)m(el)g(2)h(there)g(is)f(a)g(tile)g(of)h (lev)m(el)f(0)g(similar)150 5002 y(to)41 b(the)g(lev)m(el)g(2)g(tile,)h (that)e(this)h(rotation)f(nev)m(er)i(ends,)i(and)e(eac)m(h)g(tiling)e (con)m(tains)h(tiles)g(in)150 5122 y(in\014nitely)33 b(man)m(y)g(distinct)g(orien)m(tations.)349 5241 y(Finally)-8 b(,)55 b(for)d(an)m(y)f(in)m(tegers)i Fi(m)59 b(<)g(n)51 b Fm(w)m(e)h(consider)h(the)f(\\\()p Fi(m;)17 b(n)p Fm(\)-pin)m(wheel") 52 b(tilings)1925 5519 y(2)p eop %%Page: 3 4 3 3 bop 150 100 a Fm(de\014ned)54 b(\(for)f Fi(m)60 b Fm(=)g(2)p Fi(;)68 b(n)60 b Fm(=)g(3\))52 b(b)m(y)g(the)h(substitution) g(of)g(Figure)f(5,)57 b(whose)c(tiles)f(are)150 219 y Fi(m)7 b Fl(\000)g Fi(n)g Fl(\000)479 136 y(p)p 563 136 359 4 v 563 219 a Fi(m)650 190 y Fg(2)717 219 y Fm(+)23 b Fi(n)877 190 y Fg(2)947 219 y Fm(righ)m(t)j(triangles.)41 b(Lik)m(e)25 b(the)h(ordinary)f(pin)m(wheel,)j(suc)m(h)g(v)-6 b(arian)m(t)25 b(pin)m(wheel)150 339 y(tilings)32 b(also)h(necessarily) h(ha)m(v)m(e)g(tiles)f(in)g(in\014nitely)g(man)m(y)g(distinct)g(orien)m (tations.)349 458 y(It)h(is)h(easy)f(to)g(construct)h(explicit)e (examples)i(of)f(P)m(enrose)i(and)f(pin)m(wheel)g(tilings.)46 b(Pic)m(k)150 578 y(a)37 b(tile)f(to)h(include)h(the)f(origin)g(of)g (the)h(plane.)56 b(Em)m(b)s(ed)38 b(this)f(tile)f(in)h(a)g(tile)g(of)g (lev)m(el)g(1)g(\(there)150 697 y(are)g(sev)m(eral)h(w)m(a)m(ys)f(to)f (do)i(this\).)55 b(Em)m(b)s(ed)38 b(that)f(tile)f(of)i(lev)m(el)e(1)h (in)h(a)e(tile)h(of)g(lev)m(el)g(2,)h(em)m(b)s(ed)150 817 y(that)e(in)h(a)g(tile)f(of)h(lev)m(el)g(3,)g(and)h(so)e(on.)56 b(The)37 b(union)g(of)g(these)h(tiles)f(of)g(all)f(lev)m(els)h(will)f (co)m(v)m(er)150 936 y(an)d(in\014nite)h(region,)f(t)m(ypically|though) f(not)h(necessarily|the)h(en)m(tire)f(plane.)349 1056 y(In)d(order)g(to)g(generalize)g(from)g(the)g(ab)s(o)m(v)m(e)g (examples)g(w)m(e)g(need)h(some)f(further)h(notation.)150 1176 y(A)38 b(\\patc)m(h")g(is)g(a)g(\(\014nite)g(or)g(in\014nite\))g (subset)h(of)f(a)g(tiling)f Fi(x)e Fl(2)h Fi(X)8 b Fm(\()p Fl(A)p Fm(\);)39 b(the)f(set)g(of)g(all)g(\014nite)150 1295 y(patc)m(hes)c(for)f Fl(A)g Fm(will)e(b)s(e)i(denoted)h Fi(W)1564 1310 y Fh(A)1633 1295 y Fm(.)44 b(A)33 b(\\substitution)g (function")g Fi(\036)h Fm(is)e(a)h(map)g(from)g Fi(W)3681 1310 y Fh(A)150 1415 y Fm(to)j(itself)h(de\014ned)h(b)m(y)f(a)f(decomp) s(osition)h(of)g(eac)m(h)h(tile)d(t)m(yp)s(e,)i(stretc)m(hed)i (linearly)c(ab)s(out)i(the)150 1534 y(origin)i(b)m(y)h(a)g(factor)g Fi(\025)1024 1549 y Fj(\036)1116 1534 y Fi(>)g Fm(1,)h(in)m(to)f (congruen)m(t)h(copies)g(of)f(the)g(original)f(tiles.)64 b(\(Recall)40 b(the)150 1654 y(P)m(enrose)34 b(and)g(pin)m(wheel)g (examples)g(.\))44 b(W)-8 b(e)33 b(assume:)233 1833 y(i\))49 b(F)-8 b(or)38 b(eac)m(h)i Fi(k)f(>)e Fm(0)h(there)h(are)g(only)e(a)i (\014nite)f(n)m(um)m(b)s(er)i(of)f(p)s(ossible)g(patc)m(hes,)i(up)e(to) f(Eu-)349 1953 y(clidean)e(motion,)f(obtained)h(b)m(y)f(taking)g(a)g (ball)g(of)g(radius)h Fi(k)j Fm(around)d(an)m(y)f(p)s(oin)m(t)g(inside) 349 2072 y(a)k(tile)g(of)g(lev)m(el)g Fi(n)p Fm(,)h Fi(\036)1155 2036 y Fj(n)1210 2072 y Fm(\()p Fi(T)14 b Fm(\),)39 b(where)h Fi(T)54 b Fm(and)39 b Fi(n)g Fm(are)g(arbitrary)-8 b(.)61 b(\(This)39 b(is)g(usually)g(called)349 2192 y(\\\014nite)34 b(lo)s(cal)e(complexit)m(y".\))205 2312 y(ii\))49 b(F)-8 b(or)30 b(eac)m(h)h(tile)e Fi(T)41 b Fl(2)29 b(A)p Fm(,)g Fi(\036T)45 b Fm(con)m(tains)30 b(at)g(least)f(one)i(tile)e(of)h(eac)m (h)h(t)m(yp)s(e.)43 b(\(This)29 b(is)h(usually)349 2431 y(called)k(\\primitivit)m(y".\))178 2551 y(iii\))48 b(F)-8 b(or)28 b(ev)m(ery)g(tile)g Fi(T)42 b Fl(2)28 b(A)g Fm(there)h(is)f Fi(n)1655 2566 y Fj(T)1745 2551 y Fl(\025)g Fm(1)g(suc)m(h)i(that)e Fi(\036)2418 2515 y Fj(n)2467 2525 y Fe(T)2528 2551 y Fi(T)42 b Fm(con)m(tains)29 b(a)f(tile)f(of)i(the)g(same)349 2670 y(t)m(yp)s(e)k(as)h Fi(T)47 b Fm(and)34 b(parallel)e(to)h(it.)349 2850 y(Condition)26 b(i\))g(is)g(signi\014can)m(t,)i(the)e(remaining)g (conditions)h(m)m(uc)m(h)h(less)e(so.)42 b(In)27 b(in)m(teresting)150 2969 y(cases)j(condition)f(ii\))f(can)i(usually)f(b)s(e)g(obtained)h(b) m(y)f(replacing)g(the)h(substitution)f(b)m(y)g(a)g(p)s(o)m(w)m(er)150 3089 y(of)j(itself;)g(note)g(that)f(this)g(do)s(es)i(not)e(a\013ect)h (the)g(tiling)e(dynamical)i(system)f(at)h(all.)42 b(W)-8 b(e)32 b(kno)m(w)150 3208 y(of)h(no)h(in)m(teresting)f(examples)h(of)f (systems)h(not)f(satisfying)g(condition)g(iii\).)150 3388 y Fn(De\014nition)c(3.)43 b Fm(F)-8 b(or)27 b(a)g(giv)m(en)h (alphab)s(et)g Fl(A)f Fm(of)h(\(p)s(ossibly)f(colored\))g(tiles,)h(and) g(a)g(substitution)150 3508 y(function)i Fi(\036)p Fm(,)g(the)g (\\substitution)f(tiling)g(space")h(is)f(the)h(compact)f(subspace)i Fi(X)3143 3523 y Fj(\036)3223 3508 y Fl(\032)e Fi(X)8 b Fm(\()p Fl(A)p Fm(\),)28 b(in-)150 3627 y(v)-6 b(arian)m(t)27 b(under)i Fl(G)818 3642 y Fj(E)885 3627 y Fm(,)f(of)g(those)g(tilings)e Fi(x)i Fm(suc)m(h)h(that)d(ev)m(ery)i(\014nite)g(subpatc)m(h)h(of)e Fi(x)g Fm(is)h(congruen)m(t)150 3747 y(to)33 b(a)g(subpatc)m(h)i(of)e Fi(\036)951 3710 y Fj(n)1006 3747 y Fm(\()p Fi(T)14 b Fm(\))33 b(for)g(some)h Fi(n)27 b(>)h Fm(0)33 b(and)h Fi(T)42 b Fl(2)28 b(A)p Fm(.)349 3926 y(The)45 b(ab)s(o)m(v)m(e)e (de\014nition)i(yields)f(spaces)h(of)f(tilings)f(in)h(whic)m(h)g(the)g (tiles)g(app)s(ear)g(in)g(all)150 4046 y(orien)m(tations,)29 b(although)f(tiles)g(app)s(ear)h(in)f(at)g(most)g(coun)m(tably)h(man)m (y)f(di\013eren)m(t)h(orien)m(tations)150 4165 y(within)j(an)m(y)f Fc(\014xe)-5 b(d)32 b Fm(tiling.)43 b(An)m(y)31 b(particular)h(P)m (enrose)h(tiling,)e(for)h(example,)g(has)g(tiles)g(in)g(only)150 4285 y(10)45 b(distinct)g(orien)m(tations,)i(but)f(the)f(space)h(of)f Fc(al)5 b(l)46 b Fm(P)m(enrose)g(tilings)f(con)m(tains)g(all)f(rotated) 150 4404 y(v)m(ersions)34 b(of)f(that)g(tiling.)349 4524 y(F)-8 b(or)37 b(eac)m(h)h(tiling)e Fi(x)h Fm(and)h(eac)m(h)g Fi(r)f(>)d Fm(0,)k(consider)g(the)f(set)h(of)f(Euclidean)h(motions)f Fi(g)j Fm(for)150 4643 y(whic)m(h)30 b Fi(x)g Fm(and)g Fi(g)t(x)e Fm(agree)i(exactly)e(on)i(a)f(ball)g(of)h(radius)g Fi(r)i Fm(around)e(the)g(origin.)42 b(The)30 b(subgroup)150 4763 y(of)40 b Fi(S)6 b(O)s Fm(\()p Fi(d)p Fm(\))39 b(generated)h(b)m (y)g(the)g(rotational)e(parts)i(of)g(the)g Fi(g)t Fm('s)f(is)g(denoted) i Fl(G)3100 4778 y Fg(R)n(O)3222 4763 y Fm(\()p Fi(r)m(;)17 b(x)p Fm(\))39 b(and)h(is)150 4883 y(called)d(the)h Fc(r)-5 b(elative)39 b(orientation)f(gr)-5 b(oup)37 b Fm(of)h Fi(x)p Fm(.)56 b(In)37 b([RS1])g(it)g(w)m(as)h(sho)m(wn)g(that)f Fl(G)3303 4898 y Fg(R)n(O)3425 4883 y Fm(\()p Fi(r)m(;)17 b(x)p Fm(\))36 b(is)150 5002 y(indep)s(enden)m(t)k(of)d Fi(r)j Fm(\(and)e(w)m(e)f(henceforth)i(write)e(it)g(as)g Fl(G)2350 5017 y Fg(R)n(O)2472 5002 y Fm(\()p Fi(x)p Fm(\)\),)h(and)f(that)g(the)h(groups)g(for)150 5122 y(di\013eren)m(t)c (tilings)f Fi(x)27 b Fl(2)h Fi(X)1104 5137 y Fj(\036)1190 5122 y Fm(are)33 b(related)h(b)m(y)f(inner)h(automorphism)f(of)h Fi(S)6 b(O)s Fm(\()p Fi(d)p Fm(\).)1925 5519 y(3)p eop %%Page: 4 5 4 4 bop 960 3134 a @beginspecial 150 @llx 133 @lly 325 @urx 410 @ury 1980 @rwi @setspecial %%BeginDocument: fig4.ps %!PS-Adobe-3.0 EPSF-3.0 %%Creator: Tk Canvas Widget %%For: Charles Radin,dept,fac,000000 %%Title: Window .f3.c %%CreationDate: Fri Mar 31 14:51:47 2000 %%BoundingBox: 150 133 325 410 gsave newpath 150 133 moveto 325 133 lineto 325 410 lineto 150 410 lineto closepath clip newpath %%Pages: 1 %%DocumentData: Clean7Bit %%Orientation: Portrait %%EndComments %%BeginProlog 50 dict begin % This is a standard prolog for Postscript generated by Tk's canvas % widget. % SCCS: @(#) prolog.ps 1.5 96/02/17 17:45:11 % The definitions below just define all of the variables used in % any of the procedures here. This is needed for obscure reasons % explained on p. 716 of the Postscript manual (Section H.2.7, % "Initializing Variables," in the section on Encapsulated Postscript). /baseline 0 def /stipimage 0 def /height 0 def /justify 0 def /lineLength 0 def /spacing 0 def /stipple 0 def /strings 0 def /xoffset 0 def /yoffset 0 def /tmpstip null def % Define the array ISOLatin1Encoding (which specifies how characters are % encoded for ISO-8859-1 fonts), if it isn't already present (Postscript % level 2 is supposed to define it, but level 1 doesn't). systemdict /ISOLatin1Encoding known not { /ISOLatin1Encoding [ /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /exclam /quotedbl /numbersign /dollar /percent /ampersand /quoteright /parenleft /parenright /asterisk /plus /comma /minus /period /slash /zero /one /two /three /four /five /six /seven /eight /nine /colon /semicolon /less /equal /greater /question /at /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O /P /Q /R /S /T /U /V /W /X /Y /Z /bracketleft /backslash /bracketright /asciicircum /underscore /quoteleft /a /b /c /d /e /f /g /h /i /j /k /l /m /n /o /p /q /r /s /t /u /v /w /x /y /z /braceleft /bar /braceright /asciitilde /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /space /dotlessi /grave /acute /circumflex /tilde /macron /breve /dotaccent /dieresis /space /ring /cedilla /space /hungarumlaut /ogonek /caron /space /exclamdown /cent /sterling /currency /yen /brokenbar /section /dieresis /copyright /ordfeminine /guillemotleft /logicalnot /hyphen /registered /macron /degree /plusminus /twosuperior /threesuperior /acute /mu /paragraph /periodcentered /cedillar /onesuperior /ordmasculine /guillemotright /onequarter /onehalf /threequarters /questiondown /Agrave /Aacute /Acircumflex /Atilde /Adieresis /Aring /AE /Ccedilla /Egrave /Eacute /Ecircumflex /Edieresis /Igrave /Iacute /Icircumflex /Idieresis /Eth /Ntilde /Ograve /Oacute /Ocircumflex /Otilde /Odieresis /multiply /Oslash /Ugrave /Uacute /Ucircumflex /Udieresis /Yacute /Thorn /germandbls /agrave /aacute /acircumflex /atilde /adieresis /aring /ae /ccedilla /egrave /eacute /ecircumflex /edieresis /igrave /iacute /icircumflex /idieresis /eth /ntilde /ograve /oacute /ocircumflex /otilde /odieresis /divide /oslash /ugrave /uacute /ucircumflex /udieresis /yacute /thorn /ydieresis ] def } if % font ISOEncode font % This procedure changes the encoding of a font from the default % Postscript encoding to ISOLatin1. It's typically invoked just % before invoking "setfont". The body of this procedure comes from % Section 5.6.1 of the Postscript book. /ISOEncode { dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding ISOLatin1Encoding def currentdict end % I'm not sure why it's necessary to use "definefont" on this new % font, but it seems to be important; just use the name "Temporary" % for the font. /Temporary exch definefont } bind def % StrokeClip % % This procedure converts the current path into a clip area under % the assumption of stroking. It's a bit tricky because some Postscript % interpreters get errors during strokepath for dashed lines. If % this happens then turn off dashes and try again. /StrokeClip { {strokepath} stopped { (This Postscript printer gets limitcheck overflows when) = (stippling dashed lines; lines will be printed solid instead.) = [] 0 setdash strokepath} if clip } bind def % desiredSize EvenPixels closestSize % % The procedure below is used for stippling. Given the optimal size % of a dot in a stipple pattern in the current user coordinate system, % compute the closest size that is an exact multiple of the device's % pixel size. This allows stipple patterns to be displayed without % aliasing effects. /EvenPixels { % Compute exact number of device pixels per stipple dot. dup 0 matrix currentmatrix dtransform dup mul exch dup mul add sqrt % Round to an integer, make sure the number is at least 1, and compute % user coord distance corresponding to this. dup round dup 1 lt {pop 1} if exch div mul } bind def % width height string StippleFill -- % % Given a path already set up and a clipping region generated from % it, this procedure will fill the clipping region with a stipple % pattern. "String" contains a proper image description of the % stipple pattern and "width" and "height" give its dimensions. Each % stipple dot is assumed to be about one unit across in the current % user coordinate system. This procedure trashes the graphics state. /StippleFill { % The following code is needed to work around a NeWSprint bug. /tmpstip 1 index def % Change the scaling so that one user unit in user coordinates % corresponds to the size of one stipple dot. 1 EvenPixels dup scale % Compute the bounding box occupied by the path (which is now % the clipping region), and round the lower coordinates down % to the nearest starting point for the stipple pattern. Be % careful about negative numbers, since the rounding works % differently on them. pathbbox 4 2 roll 5 index div dup 0 lt {1 sub} if cvi 5 index mul 4 1 roll 6 index div dup 0 lt {1 sub} if cvi 6 index mul 3 2 roll % Stack now: width height string y1 y2 x1 x2 % Below is a doubly-nested for loop to iterate across this area % in units of the stipple pattern size, going up columns then % across rows, blasting out a stipple-pattern-sized rectangle at % each position 6 index exch { 2 index 5 index 3 index { % Stack now: width height string y1 y2 x y gsave 1 index exch translate 5 index 5 index true matrix tmpstip imagemask grestore } for pop } for pop pop pop pop pop } bind def % -- AdjustColor -- % Given a color value already set for output by the caller, adjusts % that value to a grayscale or mono value if requested by the CL % variable. /AdjustColor { CL 2 lt { currentgray CL 0 eq { .5 lt {0} {1} ifelse } if setgray } if } bind def % x y strings spacing xoffset yoffset justify stipple DrawText -- % This procedure does all of the real work of drawing text. The % color and font must already have been set by the caller, and the % following arguments must be on the stack: % % x, y - Coordinates at which to draw text. % strings - An array of strings, one for each line of the text item, % in order from top to bottom. % spacing - Spacing between lines. % xoffset - Horizontal offset for text bbox relative to x and y: 0 for % nw/w/sw anchor, -0.5 for n/center/s, and -1.0 for ne/e/se. % yoffset - Vertical offset for text bbox relative to x and y: 0 for % nw/n/ne anchor, +0.5 for w/center/e, and +1.0 for sw/s/se. % justify - 0 for left justification, 0.5 for center, 1 for right justify. % stipple - Boolean value indicating whether or not text is to be % drawn in stippled fashion. If text is stippled, % procedure StippleText must have been defined to call % StippleFill in the right way. % % Also, when this procedure is invoked, the color and font must already % have been set for the text. /DrawText { /stipple exch def /justify exch def /yoffset exch def /xoffset exch def /spacing exch def /strings exch def % First scan through all of the text to find the widest line. /lineLength 0 def strings { stringwidth pop dup lineLength gt {/lineLength exch def} {pop} ifelse newpath } forall % Compute the baseline offset and the actual font height. 0 0 moveto (TXygqPZ) false charpath pathbbox dup /baseline exch def exch pop exch sub /height exch def pop newpath % Translate coordinates first so that the origin is at the upper-left % corner of the text's bounding box. Remember that x and y for % positioning are still on the stack. translate lineLength xoffset mul strings length 1 sub spacing mul height add yoffset mul translate % Now use the baseline and justification information to translate so % that the origin is at the baseline and positioning point for the % first line of text. justify lineLength mul baseline neg translate % Iterate over each of the lines to output it. 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b(b)s(e)f(top)s(ologically)e (conjugate.)349 3277 y(F)-8 b(or)27 b(eac)m(h)g(tiling)f Fi(x)h Fl(2)h Fi(X)1254 3292 y Fj(\036)1307 3277 y Fm(,)g(consider)f (the)g(closure)h(of)f(the)g(orbit)f(of)h Fi(x)g Fm(under)h (translations.)150 3396 y(Let)38 b Fl(G)392 3411 y Fj(r)r(el)500 3396 y Fm(\()p Fi(x)p Fm(\))f(b)s(e)h(the)g(maximal)e(subgroup)j(of)f Fl(G)2013 3411 y Fj(E)2118 3396 y Fm(that)g(maps)g(this)f(orbit)h (closure)g(to)f(itself.)150 3516 y(It)f(is)g(not)f(hard)i(to)e(see)i (that)f Fl(G)1345 3531 y Fj(r)r(el)1452 3516 y Fm(\()p Fi(x)p Fm(\))g(is)g(the)g(smallest)g(subgroup)h(of)f Fl(G)2896 3531 y Fj(E)3000 3516 y Fm(that)f(con)m(tains)i(the)150 3636 y(closure)e(of)f Fl(G)657 3651 y Fg(R)n(O)779 3636 y Fm(\()p Fi(x)p Fm(\))g(in)g Fi(S)6 b(O)s Fm(\()p Fi(d)p Fm(\))33 b(and)h(con)m(tains)h(all)e(translations.)46 b(If)34 b(the)h(translational)d(orbit)150 3755 y(of)42 b Fi(x)f Fm(is)g(dense)i(in)f Fi(X)966 3770 y Fj(\036)1018 3755 y Fm(,)h(as)f(with)f(the)h(pin)m(wheel)g(and)g(its)f(v)-6 b(ariations,)42 b(then)g Fl(G)3193 3770 y Fj(r)r(el)3301 3755 y Fm(\()p Fi(x)p Fm(\))f(=)h Fl(G)3655 3770 y Fj(E)3722 3755 y Fm(.)150 3875 y(W)-8 b(e)39 b(should)g(add)g(that)e(for)i (tiling)e(systems)i(in)f(whic)m(h)h(tiles)f(only)g(app)s(ear)h(in)f (\014nitely)g(man)m(y)150 3994 y(orien)m(tations)j(in)g(an)m(y)f (tiling,)i(it)e(can)h(b)s(e)h(con)m(v)m(enien)m(t)g(to)e(consider)i (the)f(dynamical)g(system)150 4114 y(of)35 b(translations)g(on)h(the)f (orbit)g(closure)h(of)f(suc)m(h)i(a)e(tiling.)49 b(Although)35 b(our)g(con)m(v)m(en)m(tion)h(is)f(to)150 4233 y(allo)m(w)d(all)g(of)h (the)g(Euclidean)g(group)g(to)f(act)h(on)g(tilings)f(it)g(is)g(easy)h (to)f(obtain)h(corresp)s(onding)150 4353 y(information)g(ab)s(out)g (these)h(more)f(limited)g(dynamical)g(systems)h(from)f(our)g(results.) 349 4472 y(Note)d(that)g(the)h(map)f Fi(\036)h Fm(extends)g(naturally)f (to)f(a)i(con)m(tin)m(uous)g(map)g(\(again)e(denoted)j Fi(\036)p Fm(\))150 4592 y(from)i Fi(X)469 4607 y Fj(\036)554 4592 y Fm(in)m(to)f(itself.)44 b(Our)34 b(second)h(main)e(result)g(is:) 150 4772 y Fn(Theorem)j(2.)44 b Fm(If)33 b(for)f(a)g(substitution)h (function)g Fi(\036)f Fm(there)h(is)f(some)h(tiling)e Fi(x)h Fm(in)g Fi(X)3282 4787 y Fj(\036)3366 4772 y Fm(not)g(\014xed) 150 4891 y(b)m(y)37 b(an)m(y)g(translation,)f(for)h(whic)m(h)h(the)f (orbit)f(under)j(translation)d(is)g(dense)j(in)d Fi(X)3263 4906 y Fj(\036)3316 4891 y Fm(,)h(then)h(the)150 5011 y(extension)c(of)f Fi(\036)h Fm(to)e Fi(X)1001 5026 y Fj(\036)1087 5011 y Fm(is)h(a)g(homeomorphism.)349 5190 y(Theorem)j(2)f(complemen)m(ts)h(a)f(result)g(of)h(Solom)m(y)m(ak)e ([S],)g(whic)m(h)i(dealt)f(with)g(tiling)f(sys-)1925 5519 y(6)p eop %%Page: 7 8 7 7 bop 150 100 a Fm(tems)49 b(that)f(had)h(\014nite)g(relativ)m(e)f (orien)m(tation)g(groups.)91 b(Solom)m(y)m(ak's)48 b(result)h(w)m(as)f (itself)h(a)150 219 y(generalization)33 b(of)h(a)g(result)g(of)g(Moss)m (\023)-47 b(e)35 b([M])f(on)g(one)g(dimensional)g(subshifts.)48 b(The)34 b(in)m(terme-)150 339 y(diate)h(case)g(in)g(3)g(or)g(more)g (dimensions,)h(where)g(the)g(relativ)m(e)e(orien)m(tation)g(group)i(is) f(in\014nite)150 458 y(but)f(not)f(dense)i(in)e Fi(S)6 b(O)s Fm(\()p Fi(d)p Fm(\),)32 b(will)g(b)s(e)h(treated)h(elsewhere)g ([HRS].)150 758 y Fn(I)s(I.)39 b(Pro)s(ofs)h(of)e(Theorems)g(1)g(and)g (2,)g(and)f(some)i(further)e(results)349 937 y Fm(F)-8 b(or)24 b Fi(i)k Fm(=)g(1)p Fi(;)17 b Fm(2)p Fi(;)23 b Fm(let)h Fi(\036)1076 952 y Fj(i)1134 937 y Fm(b)s(e)h(a)f (substitution)g(on)h(alphab)s(et)f Fl(A)2493 952 y Fj(i)2550 937 y Fm(with)g(linear)g(scaling)h(factor)f Fi(\025)3689 952 y Fj(i)3722 937 y Fi(:)150 1057 y Fm(W)-8 b(rite)34 b Fi(X)511 1072 y Fj(i)579 1057 y Fm(for)g Fi(X)815 1072 y Fj(\036)863 1082 y Fe(i)934 1057 y Fm(and)h(supp)s(ose)h Fi( )d Fm(:)d(\()p Fi(X)1783 1072 y Fg(1)1827 1057 y Fi(;)17 b Fl(G)1931 1072 y Fj(E)1997 1057 y Fm(\))30 b Fl(7!)f Fm(\()p Fi(X)2317 1072 y Fg(2)2361 1057 y Fi(;)17 b Fl(G)2465 1072 y Fj(E)2532 1057 y Fm(\))34 b(is)h(a)f(top)s(ological) f(conjugacy)-8 b(.)150 1176 y(In)36 b(Theorem)h(2)f(w)m(e)h(sho)m(w)f (that)g(under)i(rather)e(general)g(circumstances)i Fi(\036)3039 1191 y Fj(i)3105 1176 y Fm(:)32 b Fi(X)3248 1191 y Fj(i)3313 1176 y Fl(7!)g Fi(X)3528 1191 y Fj(i)3597 1176 y Fm(is)k(a)150 1296 y(homeomorphism,)j(but)e(for)h(Theorem)f(1)g(w)m(e)h(simply)f (assume)h(this.)56 b(This)37 b(implies)g(that)g(no)150 1416 y(elemen)m(t)d(of)g Fi(X)715 1431 y Fj(i)780 1416 y Fm(has)g(a)f(nonzero)h(translational)e(p)s(erio)s(d.)150 1595 y Fn(Notation.)87 b Fm(F)-8 b(or)48 b Fi(r)55 b(>)e Fm(0,)e Fn(a)i Fl(2)f Fk(E)1561 1559 y Fj(d)1656 1595 y Fm(and)d(a)f(tiling)f Fi(y)t Fm(,)k Fi(B)2446 1610 y Fj(r)2490 1595 y Fm(\()p Fn(a)p Fm(\))2624 1559 y Fj(y)2719 1595 y Fm(denotes)e(the)f(patc)m(h)h(of)g Fi(y)150 1715 y Fm(consisting)c(of)g(all)f(tiles)g(in)h Fi(y)j Fm(that)c(in)m (tersect)h(the)g(op)s(en)h(ball)e(of)h(radius)g Fi(r)i Fm(ab)s(out)e Fn(a)p Fm(.)78 b(W)-8 b(e)150 1834 y(abbreviate)47 b Fi(B)730 1849 y Fj(r)773 1834 y Fm(\()p Fn(O)p Fm(\))937 1798 y Fj(y)1031 1834 y Fm(as)g Fi(B)1248 1798 y Fj(y)1243 1859 y(r)1296 1834 y Fm(.)84 b(Pic)m(k)46 b(an)h(\\inner)g(radius")g Fi(m)g Fm(suc)m(h)i(that)d(ev)m(ery)h(tile)f(in)g Fl(A)150 1954 y Fm(con)m(tains)38 b(an)f(op)s(en)g(ball)g(of)g(radius)h Fi(m)p Fm(.)56 b(F)-8 b(or)36 b(patc)m(h-v)-6 b(alued)39 b(functions)f Fi(P)s(;)17 b(Q)36 b Fm(on)h Fi(X)3379 1969 y Fg(1)3461 1954 y Fm(w)m(e)g(sa)m(y)150 2073 y Fi(P)45 b Fm(determines)33 b Fi(Q)f Fm(\(or)f Fi(Q)g Fm(is)h(determined)h(b)m(y)f Fi(P)14 b Fm(\))31 b(if)h(whenev)m(er)h Fi(x;)17 b(y)30 b Fl(2)e Fi(X)2929 2088 y Fg(1)3005 2073 y Fm(and)k Fi(P)14 b Fm(\()p Fi(x)p Fm(\))27 b(=)h Fi(P)14 b Fm(\()p Fi(y)t Fm(\))150 2193 y(w)m(e)34 b(also)e(ha)m(v)m(e)i Fi(Q)p Fm(\()p Fi(x)p Fm(\))27 b(=)h Fi(Q)p Fm(\()p Fi(y)t Fm(\))p Fi(:)349 2372 y Fm(It)33 b(follo)m(ws)g(from)h(the)f(fact)h (that)e Fi(\036)1662 2387 y Fj(i)1724 2372 y Fm(:)27 b Fi(X)1862 2387 y Fj(i)1922 2372 y Fl(7!)h Fi(X)2133 2387 y Fj(i)2198 2372 y Fm(is)34 b(a)e(homeomorphism)i(that)f(there)h (is)f(a)150 2492 y(recognizabilit)m(y)k(radius)i Fi(D)1206 2507 y Fj(i)1275 2492 y Fi(>)d Fm(0)i(suc)m(h)i(that)d(for)i Fi(x)c Fl(2)h Fi(X)2361 2507 y Fj(i)2432 2492 y Fm(the)i(patc)m(h)h Fi(B)2968 2456 y Fj(x)2963 2519 y(D)3028 2529 y Fe(i)3102 2492 y Fm(determines)h(the)150 2617 y(patc)m(h)34 b(consisting)f(of)h (tiles)f(con)m(taining)g(the)h(origin)e(in)h Fi(\036)2322 2574 y Fh(\000)p Fg(1)2322 2645 y Fj(i)2430 2617 y Fm(\()p Fi(x)p Fm(\))p Fi(:)150 2797 y Fn(Lemma)46 b(1.)66 b Fm(There)41 b(is)f(a)g(constan)m(t)h Fi(\032)e(>)h Fm(0)g(suc)m(h)i (that)d(if)i Fi(n)e Fl(2)h Fk(N)g Fm(and)h Fi(r)h(>)e(\025)3251 2760 y Fj(n)3251 2821 y Fg(2)3305 2797 y Fi(\032)g Fm(then)h(for)150 2945 y Fi(y)31 b Fl(2)d Fi(X)407 2960 y Fg(2)485 2945 y Fm(the)33 b(patc)m(h)h Fi(B)1011 2887 y Fj(\036)1059 2854 y Fd(\000)p Fe(n)1059 2911 y Fb(2)1161 2887 y Fg(\()p Fj(y)s Fg(\))1006 2979 y Fj(r)r(=)p Fg(\(2)p Fj(\025)1204 2955 y Fe(n)1204 3003 y Fb(2)1253 2979 y Fg(\))1322 2945 y Fm(is)f(determined)i(b)m(y)e(the)h(patc)m(h)g Fi(B)2608 2909 y Fj(y)2603 2970 y(r)2655 2945 y Fi(:)150 3138 y Fm(Pro)s(of.)88 b(T)-8 b(ak)m(e)48 b Fi(\032)k Fm(=)g(2)p Fi(D)1127 3153 y Fg(2)1172 3138 y Fi(=)p Fm(\()p Fi(\025)1319 3153 y Fg(2)1396 3138 y Fl(\000)32 b Fm(1\).)88 b(A)48 b(patc)m(h)h(of)f(radius)h Fi(r)i Fm(in)d Fi(y)j Fm(determines)e(a)f (patc)m(h)150 3257 y(of)40 b(radius)g(\()p Fi(r)28 b Fl(\000)f Fi(D)874 3272 y Fg(2)919 3257 y Fm(\))p Fi(=\025)1066 3272 y Fg(2)1150 3257 y Fm(in)39 b Fi(\036)1331 3215 y Fh(\000)p Fg(1)1331 3284 y(2)1439 3257 y Fm(\()p Fi(y)t Fm(\),)g(of)h(radius)g([\()p Fi(r)28 b Fl(\000)f Fi(D)2389 3272 y Fg(2)2434 3257 y Fm(\))p Fi(=\025)2581 3272 y Fg(2)2652 3257 y Fl(\000)f Fi(D)2837 3272 y Fg(2)2883 3257 y Fm(])p Fi(=\025)3019 3272 y Fg(2)3102 3257 y Fm(in)40 b Fi(\036)3284 3215 y Fh(\000)p Fg(2)3284 3284 y(2)3391 3257 y Fm(\()p Fi(y)t Fm(\),)f(and)150 3377 y Fi(r)s(=\025)306 3341 y Fj(n)306 3402 y Fg(2)391 3377 y Fl(\000)31 b Fi(D)581 3392 y Fg(2)626 3377 y Fm(\()p Fi(\025)723 3334 y Fh(\000)p Fg(1)723 3403 y(2)860 3377 y Fm(+)h Fi(\025)1027 3334 y Fh(\000)p Fg(2)1027 3403 y(2)1164 3377 y Fm(+)g Fl(\001)17 b(\001)g(\001)28 b Fm(+)k Fi(\025)1586 3334 y Fh(\000)p Fj(n)1586 3403 y Fg(2)1702 3377 y Fm(\))46 b(in)g Fi(\036)1975 3334 y Fh(\000)p Fj(n)1975 3403 y Fg(2)2092 3377 y Fm(\()p Fi(y)t Fm(\).)81 b(This)47 b(last)e(radius)i(is)f(greater)g(than)150 3496 y Fi(r)s(\025)256 3454 y Fh(\000)p Fj(n)256 3523 y Fg(2)394 3496 y Fl(\000)23 b Fi(D)576 3511 y Fg(2)621 3496 y Fi(=)p Fm(\()p Fi(\025)768 3511 y Fg(2)835 3496 y Fl(\000)f Fm(1\),)33 b(whic)m(h)h(in)f(turn)h(is)f(at)g(least)g Fi(r)s(=)p Fm(2)p Fi(\025)2368 3460 y Fj(n)2368 3521 y Fg(2)2421 3496 y Fm(.)p 2509 3506 60 80 v 349 3676 a(An)m(y)i(elemen)m(t)g Fi(g)e Fl(2)d(G)1169 3691 y Fj(E)1271 3676 y Fm(can)35 b(b)s(e)g(written)f(uniquely)g(as)h(the)g(comp)s (osition)f(of)h(a)f(rotation)150 3796 y(and)g(a)f(translation,)f(i.e.,) g(there)i(exist)f(unique)h Fi(\013)27 b Fl(2)h Fi(S)6 b(O)s Fm(\()p Fi(d)p Fm(\))33 b(and)g Fn(s)28 b Fl(2)g Fk(R)2888 3759 y Fj(d)2968 3796 y Fm(suc)m(h)35 b(that)1380 4022 y Fi(g)t Fn(a)26 b Fm(=)i Fi(\013)p Fn(a)22 b Fm(+)h Fn(s)33 b Fm(for)h(all)e Fn(a)27 b Fl(2)i Fk(E)2474 3981 y Fj(d)3623 4022 y Fm(\(3\))150 4249 y(and)34 b(w)m(e)f(set)1296 4369 y Fi(`)p Fm(\()p Fi(g)t Fm(\))26 b(=)i Fl(k)p Fi(\013)22 b Fl(\000)h Fi(I)8 b Fl(k)1937 4396 y Fm(op)s(erator)2332 4369 y(+)22 b Fl(k)p Fn(s)p Fl(k)p Fi(:)1019 b Fm(\(4\))150 4613 y Fn(Notation.)42 b Fm(F)-8 b(or)32 b(patc)m(h-v)-6 b(alued)33 b(functions)h Fi(P)s(;)17 b(Q)30 b Fm(on)j Fi(X)2304 4628 y Fg(1)2380 4613 y Fm(the)f(phrase)i(\\)p Fi(P)45 b Fm(determines)33 b Fi(Q)f Fm(up)150 4732 y(to)39 b(motion)f(b)m(y)h(some)g Fi(g)i Fl(2)c(G)1271 4747 y Fj(E)1378 4732 y Fm(with)i Fi(`)p Fm(\()p Fi(g)t Fm(\))c Fl(\024)j Fi(\021)t Fm(")h(means)g(that)g(if)g Fi(x;)17 b(y)40 b Fl(2)e Fi(X)3082 4747 y Fg(1)3165 4732 y Fm(are)h(suc)m(h)h (that)150 4852 y Fi(P)14 b Fm(\()p Fi(x)p Fm(\))27 b(=)h Fi(P)14 b Fm(\()p Fi(y)t Fm(\))32 b(then)i(there)g(exists)f Fi(g)e Fl(2)d(G)1725 4867 y Fj(E)1825 4852 y Fm(with)33 b Fi(`)p Fm(\()p Fi(g)t Fm(\))26 b Fl(\024)i Fi(\021)38 b Fm(suc)m(h)c(that)f Fi(Q)p Fm(\()p Fi(x)p Fm(\))27 b(=)h Fi(g)t(Q)p Fm(\()p Fi(y)t Fm(\))p Fi(:)150 5031 y Fn(Lemma)41 b(2.)51 b Fm(There)37 b(exist)e(a)g(constan)m(t)h Fi(S)1778 5046 y Fg(0)1854 5031 y Fi(>)c Fm(0)k(and)g(a)f(function)h Fi(\021)g Fm(:)31 b Fk(R)2938 5046 y Fg(+)3036 5031 y Fl(7!)g Fk(R)3239 5046 y Fg(+)3340 5031 y Fm(suc)m(h)37 b(that)150 5151 y(lim)288 5166 y Fj(r)r Fh(!1)508 5151 y Fi(\021)t Fm(\()p Fi(r)s Fm(\))27 b(=)i(0)k(and)h(if)g Fi(r)c(>)f(S)1431 5166 y Fg(0)1509 5151 y Fm(then)34 b(for)g Fi(y)e Fl(2)c Fi(X)2147 5166 y Fg(1)2225 5151 y Fm(the)34 b(patc)m(h)g Fi(B)2752 5115 y Fj(y)2747 5176 y(r)2833 5151 y Fm(determines)g(the)g(patc)m(h)150 5280 y Fi(B)231 5228 y Fj( )s Fg(\()p Fj(y)s Fg(\))226 5309 y Fj(r)r Fh(\000)p Fj(S)376 5319 y Fb(0)453 5280 y Fm(up)g(to)f(motion) f(b)m(y)i(some)f Fi(g)e Fl(2)d(G)1676 5295 y Fj(E)1777 5280 y Fm(with)33 b Fi(`)p Fm(\()p Fi(g)t Fm(\))26 b Fl(\024)i Fi(\021)t Fm(\()p Fi(r)s Fm(\))p Fi(:)1925 5519 y Fm(7)p eop %%Page: 8 9 8 8 bop 150 100 a Fm(Pro)s(of.)63 b(By)39 b(uniform)i(con)m(tin)m(uit)m (y)f(of)g Fi( )t Fm(,)g(there)h(is)f(a)f(radius)i Fi(S)2585 115 y Fg(0)2669 100 y Fm(suc)m(h)g(that)f(the)g(patc)m(h)h Fi(B)3663 52 y Fj(y)3658 129 y(S)3707 139 y Fb(0)150 219 y Fm(determines)32 b(the)f(tile)f(at)g(the)h(origin)f(of)h Fi( )t Fm(\()p Fi(y)t Fm(\))e(and)i(its)g(immediate)f(neigh)m(b)s(ors,) i(up)f(to)g(motion)150 339 y(b)m(y)j(less)g(than)g Fi(m=)p Fm(2.)46 b(Since)34 b Fi( )j Fm(is)d(a)f(conjugacy)-8 b(,)34 b(for)g(an)m(y)g(p)s(oin)m(t)g Fn(a)28 b Fl(2)h Fk(E)2846 303 y Fj(d)2893 339 y Fm(,)k(the)h(patc)m(h)h Fi(B)3477 354 y Fj(S)3526 364 y Fb(0)3569 339 y Fm(\()p Fn(a)p Fm(\))3703 303 y Fj(y)150 458 y Fm(determines)41 b(the)f(tile)f(at)g Fn(a)h Fm(in)f Fi( )t Fm(\()p Fi(y)t Fm(\))g(and)h(its)f(nearest)i(neigh)m(b)s(ors,)h(up)e(to)f(a)h(small)f (motion.)150 587 y(Applying)27 b(this)h(to)g(all)f(p)s(oin)m(ts)h Fn(a)g Fl(2)g Fi(B)1563 602 y Fj(r)r Fh(\000)p Fj(S)1713 612 y Fb(0)1756 587 y Fm(,)h(w)m(e)g(ha)m(v)m(e)f(that)g(the)g(patc)m (h)h Fi(B)2906 551 y Fj(y)2901 612 y(r)2981 587 y Fm(determines)g Fi(B)3561 535 y Fj( )s Fg(\()p Fj(y)s Fg(\))3556 617 y Fj(r)r Fh(\000)p Fj(S)3706 627 y Fb(0)150 707 y Fm(up)g(to)e(an)h(o)m (v)m(erall)f(small)h(motion)f Fi(g)t Fm(.)41 b(The)28 b(b)s(ound)h(on)f Fi(`)p Fm(\()p Fi(g)t Fm(\))e(follo)m(ws)i(from)g (uniform)h(con)m(tin)m(uit)m(y)150 826 y(of)k Fi( )t Fm(.)p 419 836 60 80 v 349 1006 a(F)-8 b(or)33 b Fi(n)27 b Fl(2)i Fk(N)k Fm(put)1563 1170 y Fi(k)s Fm(\()p Fi(n)p Fm(\))27 b(=)1889 1029 y Fa(\026)1947 1170 y Fi(n)2019 1102 y Fm(log)17 b Fi(\025)2222 1117 y Fg(1)p 2018 1147 249 4 v 2018 1238 a Fm(log)h Fi(\025)2222 1253 y Fg(2)2278 1029 y Fa(\027)3623 1170 y Fm(\(5\))150 1390 y(and)34 b(set)1484 1509 y Fi( )1549 1524 y Fj(n)1631 1509 y Fm(=)28 b Fi(\036)1795 1458 y Fh(\000)p Fj(k)r Fg(\()p Fj(n)p Fg(\))1795 1536 y(2)2040 1509 y Fl(\016)22 b Fi( )k Fl(\016)c Fi(\036)2334 1468 y Fj(n)2334 1534 y Fg(1)2388 1509 y Fi(:)1207 b Fm(\(6\))349 1696 y(Fix)34 b Fi(x)c Fl(2)h Fi(X)796 1711 y Fg(1)840 1696 y Fi(:)k Fm(Note)f(that)h Fi(\025)1421 1655 y Fj(n)1421 1721 y Fg(1)1475 1696 y Fi(=\025)1583 1645 y Fj(k)r Fg(\()p Fj(n)p Fg(\))1583 1723 y(2)1774 1696 y Fl(2)30 b Fm([1)p Fi(;)17 b(\025)2051 1711 y Fg(2)2095 1696 y Fm(\),)34 b(and)i(if)f(log)17 b Fi(\025)2688 1711 y Fg(1)2733 1696 y Fi(=)g Fm(log)f Fi(\025)3002 1711 y Fg(2)3082 1696 y Fm(is)35 b(rational)e(then)150 1827 y(the)j(sequence)h(\()p Fi(\025)836 1791 y Fj(n)836 1852 y Fg(1)890 1827 y Fi(\025)948 1775 y Fh(\000)p Fj(k)r Fg(\()p Fj(n)p Fg(\))948 1854 y(2)1171 1827 y Fm(\))1210 1791 y Fh(1)1210 1852 y Fj(n)p Fg(=1)1400 1827 y Fm(is)e(p)s(erio)s (dic)h(and)g(1)f(is)g(in)h(its)f(range,)h(while)f(if)g(log)18 b Fi(\025)3391 1842 y Fg(1)3435 1827 y Fi(=)f Fm(log)g Fi(\025)3705 1842 y Fg(2)150 1947 y Fm(is)36 b(irrational)e(then)i(the) h(range)f(of)g(this)f(sequence)j(is)e(dense)h(in)f([1)p Fi(;)17 b(\025)2776 1962 y Fg(2)2819 1947 y Fm(])p Fi(:)35 b Fm(Th)m(us)i(there)g(exists)e(a)150 2066 y(subsequence)h Fi(n)773 2081 y Fj(i)839 2066 y Fm(suc)m(h)f(that)e Fi( )1345 2081 y Fj(n)1394 2091 y Fe(i)1430 2066 y Fm(\()p Fi(x)p Fm(\))g(con)m(v)m(erges,)h(sa)m(y)f(to)g Fi(x)2423 2030 y Fh(0)2478 2066 y Fl(2)c Fi(X)2656 2081 y Fg(2)2700 2066 y Fi(;)k Fm(and)g(suc)m(h)i(that)1300 2284 y Fi(\025)1358 2233 y Fh(\000)p Fj(k)r Fg(\()p Fj(n)1544 2243 y Fe(i)1577 2233 y Fg(\))1358 2311 y(2)1613 2284 y Fi(\025)1671 2240 y Fj(n)1720 2250 y Fe(i)1671 2311 y Fg(1)1784 2284 y Fl(!)28 b Fm(1)199 b(as)33 b Fi(i)28 b Fl(!)f(1)p Fi(:)1023 b Fm(\(7\))150 2493 y Fn(Theorem)56 b(3.)93 b Fm(The)50 b(sequence)h Fl(f)p Fi( )1602 2508 y Fj(n)1651 2518 y Fe(i)1687 2493 y Fl(g)1737 2457 y Fh(1)1737 2519 y Fj(i)p Fg(=1)1920 2493 y Fm(con)m(v)m(erges)g(uniformly)e(to)g(a)g(conjugacy)h Fi( )3640 2457 y Fh(0)3722 2493 y Fm(:)150 2613 y(\()p Fi(X)272 2628 y Fg(1)316 2613 y Fi(;)17 b Fl(G)420 2628 y Fj(E)487 2613 y Fm(\))50 b Fl(7!)g Fm(\()p Fi(X)848 2628 y Fg(2)892 2613 y Fi(;)17 b Fl(G)996 2628 y Fj(E)1062 2613 y Fm(\))47 b(suc)m(h)h(that,)i(for)d Fi(r)53 b(>)d(\032)31 b Fm(+)h Fi(S)2291 2628 y Fg(0)2382 2613 y Fm(and)48 b(for)f Fi(y)53 b Fl(2)e Fi(X)3058 2628 y Fg(1)3149 2613 y Fm(the)c(patc)m(h)h Fi(B)3703 2577 y Fj(y)3698 2637 y(r)150 2747 y Fm(determines)34 b(the)g(patc)m(h)g Fi(B)1181 2695 y Fj( )1237 2665 y Fd(0)1263 2695 y Fg(\()p Fj(y)s Fg(\))1176 2781 y Fj(r)r(=)p Fg(2)1373 2747 y Fi(:)150 2933 y Fm(Pro)s(of.)82 b(Step)47 b(1.)82 b(If)47 b Fi(g)52 b Fl(2)d(G)1287 2948 y Fj(E)1401 2933 y Fm(then)e Fi( )1706 2948 y Fj(n)1755 2958 y Fe(i)1791 2933 y Fm(\()p Fi(g)t(x)p Fm(\))h Fl(!)h Fi(g)t(x)2284 2897 y Fh(0)2357 2933 y Fm(as)d Fi(i)j Fl(!)g(1)p Fi(:)c Fm(Indeed,)51 b(if)46 b(w)m(e)g(write)150 3053 y Fi(g)t(x)27 b Fm(=)h Fi(\013x)22 b Fm(+)h Fn(s)33 b Fm(as)g(in)h(equation)e(\(3\))h(then)1073 3271 y Fi( )1138 3286 y Fj(n)1187 3296 y Fe(i)1223 3271 y Fm(\()p Fi(g)t(x)p Fm(\))26 b(=)j(\()p Fi(\036)1640 3220 y Fh(\000)p Fj(k)r Fg(\()p Fj(n)1826 3230 y Fe(i)1858 3220 y Fg(\))1640 3298 y(2)1916 3271 y Fl(\016)22 b Fi( )t Fm(\)\()p Fi(\036)2194 3227 y Fj(n)2243 3237 y Fe(i)2194 3298 y Fg(1)2279 3271 y Fm(\()p Fi(g)t(x)p Fm(\)\))1436 3440 y(=)29 b(\()p Fi(\036)1640 3388 y Fh(\000)p Fj(k)r Fg(\()p Fj(n)1826 3398 y Fe(i)1858 3388 y Fg(\))1640 3467 y(2)1916 3440 y Fl(\016)22 b Fi( )t Fm(\)\()p Fi(\013\036)2258 3396 y Fj(n)2307 3406 y Fe(i)2258 3467 y Fg(1)2343 3440 y Fm(\()p Fi(x)p Fm(\))g(+)h Fi(\025)2658 3396 y Fj(n)2707 3406 y Fe(i)2658 3467 y Fg(1)2743 3440 y Fn(s)p Fm(\))1436 3609 y(=)29 b Fi(\036)1601 3557 y Fh(\000)p Fj(k)r Fg(\()p Fj(n)1787 3567 y Fe(i)1819 3557 y Fg(\))1601 3635 y(2)1855 3609 y Fm(\()p Fi(\013 )t Fm([)p Fi(\036)2114 3565 y Fj(n)2163 3575 y Fe(i)2114 3635 y Fg(1)2199 3609 y Fm(\()p Fi(x)p Fm(\)])22 b(+)g Fi(\025)2541 3565 y Fj(n)2590 3575 y Fe(i)2541 3635 y Fg(1)2627 3609 y Fn(s)p Fm(\))1436 3778 y(=)29 b Fi(\013 )1671 3793 y Fj(n)1720 3803 y Fe(i)1756 3778 y Fm(\()p Fi(x)p Fm(\))22 b(+)h Fi(\025)2071 3726 y Fh(\000)p Fj(k)r Fg(\()p Fj(n)2257 3736 y Fe(i)2289 3726 y Fg(\))2071 3804 y(2)2325 3778 y Fi(\025)2383 3733 y Fj(n)2432 3743 y Fe(i)2383 3804 y Fg(1)2469 3778 y Fn(s)3623 3515 y Fm(\(8\))150 3977 y(whic)m(h)34 b(clearly)f(con)m(v)m (erges)h(to)f Fi(g)t(x)1431 3941 y Fh(0)1491 3977 y Fm(as)g Fi(i)28 b Fl(!)g(1)p Fi(:)150 4156 y Fm(Step)34 b(2.)44 b(Supp)s(ose)34 b Fi(g)t(;)17 b(g)1038 4120 y Fh(0)1092 4156 y Fl(2)28 b(G)1245 4171 y Fj(E)1345 4156 y Fm(and)34 b Fi(r)d(>)d(\032)21 b Fm(+)i Fi(S)1954 4171 y Fg(0)2032 4156 y Fm(are)33 b(suc)m(h)i(that)e Fi(B)2720 4120 y Fj(g)s(x)2715 4181 y(r)2839 4156 y Fm(=)28 b Fi(B)3025 4120 y Fj(g)3067 4090 y Fd(0)3093 4120 y Fj(x)3020 4181 y(r)3143 4156 y Fm(.)44 b(F)-8 b(or)33 b(eac)m(h)h Fi(i;)1494 4390 y(B)1575 4332 y Fj(\036)1623 4292 y Fe(n)1667 4308 y(i)1623 4356 y Fb(1)1702 4332 y Fg(\()p Fj(g)s(x)p Fg(\))1570 4433 y Fj(\025)1617 4393 y Fe(n)1661 4409 y(i)1617 4457 y Fb(1)1697 4433 y Fj(r)1884 4390 y Fm(=)28 b Fi(B)2070 4332 y Fj(\036)2118 4292 y Fe(n)2162 4308 y(i)2118 4356 y Fb(1)2198 4332 y Fg(\()p Fj(g)2271 4302 y Fd(0)2297 4332 y Fj(x)p Fg(\))2065 4433 y Fj(\025)2112 4393 y Fe(n)2156 4409 y(i)2112 4457 y Fb(1)2192 4433 y Fj(r)2379 4390 y Fi(:)1216 b Fm(\(9\))150 4621 y(By)32 b(Lemma)i(2,)f(there)h(exists)f Fi(h)1360 4636 y Fj(i)1421 4621 y Fl(2)28 b(G)1574 4636 y Fj(E)1675 4621 y Fm(with)33 b Fi(`)p Fm(\()p Fi(h)2040 4636 y Fj(i)2072 4621 y Fm(\))28 b Fl(\024)g Fi(\021)t Fm(\()p Fi(\025)2394 4577 y Fj(n)2443 4587 y Fe(i)2394 4648 y Fg(1)2480 4621 y Fi(r)s Fm(\))k(suc)m(h)j(that)1354 4857 y Fi(B)1435 4799 y Fj(h)1482 4809 y Fe(i)1513 4799 y Fj( )s Fg([)p Fj(\036)1640 4759 y Fe(n)1684 4775 y(i)1640 4823 y Fb(1)1719 4799 y Fg(\()p Fj(g)s(x)p Fg(\)])1430 4900 y Fj(\025)1477 4860 y Fe(n)1521 4876 y(i)1477 4924 y Fb(1)1557 4900 y Fj(r)r Fh(\000)p Fj(S)1707 4910 y Fb(0)1923 4857 y Fm(=)28 b Fi(B)2109 4799 y Fj( )s Fg([)p Fj(\036)2236 4759 y Fe(n)2280 4775 y(i)2236 4823 y Fb(1)2315 4799 y Fg(\()p Fj(g)2388 4769 y Fd(0)2414 4799 y Fj(x)p Fg(\)])2104 4900 y Fj(\025)2151 4860 y Fe(n)2195 4876 y(i)2151 4924 y Fb(1)2231 4900 y Fj(r)r Fh(\000)p Fj(S)2381 4910 y Fb(0)2518 4857 y Fi(:)1027 b Fm(\(10\))150 5082 y(By)32 b(Lemma)i(1)1017 5237 y Fi(B)1098 5178 y Fj(\036)1146 5138 y Fd(\000)p Fe(k)q Fb(\()p Fe(n)1310 5154 y(i)1340 5138 y Fb(\))1146 5202 y(2)1372 5178 y Fg(\()p Fj(h)1450 5188 y Fe(i)1482 5178 y Fj( )s Fg([)p Fj(\036)1609 5138 y Fe(n)1653 5154 y(i)1609 5202 y Fb(1)1688 5178 y Fg(\()p Fj(g)s(x)p Fg(\)]\))1093 5296 y(\()p Fj(\025)1171 5256 y Fe(n)1215 5272 y(i)1171 5320 y Fb(1)1251 5296 y Fj(r)r Fh(\000)p Fj(S)1401 5306 y Fb(0)1441 5296 y Fg(\))p Fj(=)p Fg(2)p Fj(\025)1600 5256 y Fe(k)q Fb(\()p Fe(n)1710 5272 y(i)1741 5256 y Fb(\))1600 5320 y(2)1923 5237 y Fm(=)28 b Fi(B)2109 5178 y Fj(\036)2157 5138 y Fd(\000)p Fe(k)q Fb(\()p Fe(n)2321 5154 y(i)2352 5138 y Fb(\))2157 5202 y(2)2384 5178 y Fg(\()p Fj( )s Fg([)p Fj(\036)2542 5138 y Fe(n)2586 5154 y(i)2542 5202 y Fb(1)2621 5178 y Fg(\()p Fj(g)2694 5148 y Fd(0)2720 5178 y Fj(x)p Fg(\)]\))2104 5296 y(\()p Fj(\025)2182 5256 y Fe(n)2226 5272 y(i)2182 5320 y Fb(1)2263 5296 y Fj(r)r Fh(\000)p Fj(S)2413 5306 y Fb(0)2452 5296 y Fg(\))p Fj(=)p Fg(2)p Fj(\025)2611 5256 y Fe(k)q Fb(\()p Fe(n)2721 5272 y(i)2752 5256 y Fb(\))2611 5320 y(2)2855 5237 y Fi(:)690 b Fm(\(11\))1925 5519 y(8)p eop %%Page: 9 10 9 9 bop 349 100 a Fm(Let)34 b Fi(\013;)17 b(\014)693 115 y Fj(i)754 100 y Fl(2)29 b Fi(S)6 b(O)s Fm(\()p Fi(d)p Fm(\))32 b(and)i Fn(s)p Fi(;)17 b Fn(t)1486 115 y Fj(i)1546 100 y Fl(2)29 b Fk(R)1713 63 y Fj(d)1793 100 y Fm(b)s(e)34 b(suc)m(h)h(that)e Fi(g)t Fn(a)27 b Fm(=)h Fi(\013)p Fn(a)22 b Fm(+)h Fn(s)34 b Fm(and)g Fi(h)3182 115 y Fj(i)3215 100 y Fn(a)28 b Fm(=)h Fi(\014)3461 115 y Fj(i)3494 100 y Fn(a)22 b Fm(+)h Fn(t)3717 115 y Fj(i)150 219 y Fm(for)33 b(all)g Fn(a)27 b Fl(2)h Fk(E)684 183 y Fj(d)731 219 y Fm(.)44 b(W)-8 b(e)34 b(ha)m(v)m(e)570 450 y Fi(\036)629 398 y Fh(\000)p Fj(k)r Fg(\()p Fj(n)815 408 y Fe(i)847 398 y Fg(\))629 476 y(2)884 450 y Fm(\()p Fi(h)980 465 y Fj(i)1013 450 y Fi( )t Fm([)p Fi(\036)1169 406 y Fj(n)1218 416 y Fe(i)1169 476 y Fg(1)1254 450 y Fm(\()p Fi(g)t(x)p Fm(\)]\))26 b(=)i Fi(\014)1695 465 y Fj(i)1728 450 y Fi(\013 )1857 465 y Fj(n)1906 475 y Fe(i)1943 450 y Fm(\()p Fi(x)p Fm(\))22 b(+)g Fi(\025)2257 398 y Fh(\000)p Fj(k)r Fg(\()p Fj(n)2443 408 y Fe(i)2476 398 y Fg(\))2257 476 y(2)2512 450 y Fi(\025)2570 406 y Fj(n)2619 416 y Fe(i)2570 476 y Fg(1)2656 450 y Fi(\014)2712 465 y Fj(i)2745 450 y Fn(s)h Fm(+)f Fi(\025)2970 398 y Fh(\000)p Fj(k)r Fg(\()p Fj(n)3156 408 y Fe(i)3189 398 y Fg(\))2970 476 y(2)3225 450 y Fn(t)3270 465 y Fj(i)3302 450 y Fi(;)243 b Fm(\(12\))150 681 y(and)34 b(this)f(con)m(v)m(erges)h(to)f Fi(g)t(x)1215 644 y Fh(0)1275 681 y Fm(as)h Fi(i)28 b Fl(!)f(1)p Fi(:)32 b Fm(W)-8 b(e)34 b(kno)m(w)f(from)h(Step)g(1)f(that)1118 915 y Fi(\036)1177 863 y Fh(\000)p Fj(k)r Fg(\()p Fj(n)1363 873 y Fe(i)1395 863 y Fg(\))1177 942 y(2)1431 915 y Fm(\()p Fi( )t Fm([)p Fi(\036)1626 871 y Fj(n)1675 881 y Fe(i)1626 942 y Fg(1)1711 915 y Fm(\()p Fi(g)1802 874 y Fh(0)1829 915 y Fi(x)p Fm(\)]\))27 b Fl(!)g Fi(g)2198 874 y Fh(0)2225 915 y Fi(x)2282 874 y Fh(0)2343 915 y Fm(as)34 b Fi(i)28 b Fl(!)f(1)p Fi(:)790 b Fm(\(13\))150 1146 y(Since)1290 1220 y Fi(\025)1348 1176 y Fj(n)1397 1186 y Fe(i)1348 1247 y Fg(1)1434 1220 y Fi(r)25 b Fl(\000)e Fi(S)1665 1235 y Fg(0)p 1290 1265 420 4 v 1350 1381 a Fm(2)p Fi(\025)1458 1329 y Fj(k)r Fg(\()p Fj(n)1582 1339 y Fe(i)1614 1329 y Fg(\))1458 1407 y(2)1749 1288 y Fl(!)1889 1220 y Fi(r)p 1888 1265 50 4 v 1888 1356 a Fm(2)2183 1288 y(as)33 b Fi(i)28 b Fl(!)f(1)p Fi(;)951 b Fm(\(14\))150 1542 y(it)33 b(follo)m(ws)g(that)1657 1662 y Fi(B)1738 1614 y Fj(g)s(x)1825 1584 y Fd(0)1733 1696 y Fj(r)r(=)p Fg(2)1885 1662 y Fm(=)28 b Fi(B)2071 1614 y Fj(g)2113 1584 y Fd(0)2139 1614 y Fj(x)2184 1584 y Fd(0)2066 1696 y Fj(r)r(=)p Fg(2)2216 1662 y Fi(:)1329 b Fm(\(15\))150 1914 y(Step)28 b(3.)42 b(De\014ne)29 b Fi( )863 1878 y Fh(0)918 1914 y Fm(:)e Fi(X)1056 1929 y Fg(1)1128 1914 y Fl(7!)g Fi(X)1338 1929 y Fg(2)1410 1914 y Fm(as)h(follo)m(ws.)42 b(Giv)m(en)27 b Fi(y)k Fl(2)d Fi(X)2430 1929 y Fg(1)2502 1914 y Fm(let)f Fl(f)p Fi(g)2738 1929 y Fj(j)2780 1914 y Fl(g)2830 1929 y Fj(j)2899 1914 y Fm(b)s(e)h(a)f(sequence)i(in)e Fl(G)3682 1929 y Fj(E)150 2034 y Fm(suc)m(h)k(that)e Fi(g)631 2049 y Fj(j)673 2034 y Fi(x)f Fl(7!)f Fi(y)33 b Fm(as)d Fi(j)j Fl(!)27 b(1)p Fi(:)i Fm(There)i(exist)e(group)h(elemen)m(ts)h Fi(h)2701 2049 y Fj(j)2773 2034 y Fm(tending)f(to)f(the)h(iden)m(tit)m (y)150 2165 y(in)46 b Fl(G)338 2180 y Fj(E)452 2165 y Fm(and)h(p)s(ositiv)m(e)f(real)g(n)m(um)m(b)s(ers)i Fi(r)1708 2180 y Fj(j)1797 2165 y Fm(tending)f(to)f(in\014nit)m(y)g(suc)m(h)i (that)e Fi(B)3209 2111 y Fj(h)3256 2121 y Fe(j)3292 2111 y Fj(g)3331 2121 y Fe(j)3368 2111 y Fj(x)3204 2176 y(r)3241 2186 y Fe(j)3468 2165 y Fm(=)k Fi(B)3676 2129 y Fj(y)3671 2189 y(r)3708 2199 y Fe(j)150 2284 y Fm(for)44 b(eac)m(h)h Fi(j:)e Fm(By)g(Step)i(2,)h(\()p Fi(h)1285 2299 y Fj(j)1327 2284 y Fi(g)1375 2299 y Fj(j)1416 2284 y Fi(x)1473 2248 y Fh(0)1501 2284 y Fm(\))1540 2299 y Fj(j)1626 2284 y Fm(con)m(v)m(erges,)h(hence)e(so)f(do)s(es)h(\()p Fi(g)2856 2299 y Fj(j)2897 2284 y Fi(x)2954 2248 y Fh(0)2982 2284 y Fm(\))3021 2299 y Fj(j)3062 2284 y Fi(;)f Fm(and)g(w)m(e)g(de\014ne) 150 2404 y Fi( )219 2368 y Fh(0)246 2404 y Fm(\()p Fi(y)t Fm(\))50 b(=)h(lim)693 2419 y Fj(j)t Fh(!1)910 2404 y Fi(g)958 2419 y Fj(j)1000 2404 y Fi(x:)46 b Fm(The)i(existence)f(of)h (this)e(limit)g(ensures)j(that)d Fi( )3036 2368 y Fh(0)3111 2404 y Fm(is)g(w)m(ell)h(de\014ned)150 2523 y(and)38 b(con)m(tin)m(uous.)59 b(W)-8 b(e)38 b(see)g(from)g(Step)g(1)g(that)f (it)g(is)h(a)f(conjugacy)-8 b(,)39 b(and)f(the)g(Little)f(Caesar)150 2643 y(prop)s(ert)m(y)c(follo)m(ws)g(from)h(Step)g(2.)150 2822 y(Step)29 b(4.)42 b(W)-8 b(e)29 b(still)e(need)j(to)e(sho)m(w)h (that)f Fi( )1711 2837 y Fj(n)1760 2847 y Fe(i)1825 2822 y Fm(con)m(v)m(erges)i(to)e Fi( )2453 2786 y Fh(0)2508 2822 y Fm(uniformly)h(on)f Fi(X)3174 2837 y Fg(1)3219 2822 y Fi(:)g Fm(F)-8 b(or)28 b Fi(y)j Fl(2)d Fi(X)3706 2837 y Fg(1)150 2942 y Fm(and)g Fi(r)i(>)e Fm(0)f(one)h(can)f(\014nd)h Fi(g)j Fl(2)d(G)1373 2957 y Fj(E)1468 2942 y Fm(suc)m(h)h(that)d Fi(g)t(x)g Fm(agrees)i(with)f Fi(y)j Fm(on)e(a)f(ball)f(of)i(radius)f Fi(r)j Fm(ab)s(out)150 3062 y(the)42 b(origin.)67 b(By)40 b(linear)h(rep)s(etitivit)m(y)e(\(see)j(Lemma)g(4,)g(b)s(elo)m(w\),)h (w)m(e)f(can)f(alw)m(a)m(ys)g(c)m(ho)s(ose)h Fi(g)150 3181 y Fm(suc)m(h)35 b(that)e Fi(`)p Fm(\()p Fi(g)t Fm(\))25 b Fi(<)j(C)7 b(r)37 b Fm(for)c(some)h(\014xed)g(constan)m(t)f Fi(C)7 b Fm(.)45 b(By)32 b(the)i(triangle)e(inequalit)m(y)-8 b(,)350 3402 y Fi(m)p Fm([)p Fi( )530 3417 y Fj(n)579 3427 y Fe(i)616 3402 y Fm(\()p Fi(y)t Fm(\))p Fi(;)17 b( )861 3361 y Fh(0)887 3402 y Fm(\()p Fi(y)t Fm(\)])26 b Fl(\024)i Fi(m)p Fm([)p Fi( )1357 3417 y Fj(n)1406 3427 y Fe(i)1443 3402 y Fm(\()p Fi(y)t Fm(\))p Fi(;)o( )1666 3417 y Fj(n)1715 3427 y Fe(i)1751 3402 y Fm(\()p Fi(g)t(x)p Fm(\)])21 b(+)h Fi(m)p Fm([)p Fi( )2266 3417 y Fj(n)2315 3427 y Fe(i)2352 3402 y Fm(\()p Fi(g)t(x)p Fm(\))p Fi(;)17 b(g)t( )2701 3417 y Fj(n)2750 3427 y Fe(i)2784 3402 y Fm(\()p Fi(x)p Fm(\)])1623 3551 y(+)22 b Fi(m)p Fm([)p Fi(g)t( )1954 3566 y Fj(n)2003 3576 y Fe(i)2039 3551 y Fm(\()p Fi(x)p Fm(\))p Fi(;)17 b( )2288 3510 y Fh(0)2314 3551 y Fm(\()p Fi(g)t(x)p Fm(\)])k(+)h Fi(m)p Fm([)p Fi( )2833 3510 y Fh(0)2861 3551 y Fm(\()p Fi(g)t(x)p Fm(\))p Fi(;)17 b( )3162 3510 y Fh(0)3187 3551 y Fm(\()p Fi(y)t Fm(\)])p Fi(:)3573 3477 y Fm(\(16\))150 3772 y(Giv)m(en)29 b Fi(\017)p Fm(,)h(w)m(e)f(will)f(sho)m(w)h(that)f(for)h Fi(i)g Fm(large)g(\(with)f(estimates)h(indep)s(enden)m(t)i(of)e Fi(y)t Fm(\),)f(and)i(for)f(the)150 3892 y(correct)38 b(c)m(hoice)h(of)e Fi(g)t Fm(,)h(eac)m(h)g(term)g(on)g(the)g(righ)m(t)f (hand)i(side)f(is)g(b)s(ounded)h(b)m(y)e Fi(\017=)p Fm(4.)58 b(The)38 b(left)150 4011 y(hand)e(side)f(\(whic)m(h)h(is)e(indep)s (enden)m(t)k(of)d(the)g(c)m(hoices)h(made\))f(is)g(then)g(b)s(ounded)i (b)m(y)e Fi(\017)p Fm(.)49 b(Since)150 4131 y(the)34 b(estimates)f(on)h Fi(i)g Fm(w)m(ere)g(indep)s(enden)m(t)i(of)e Fi(y)t Fm(,)e(the)i(left)g(hand)g(side)g(go)s(es)g(to)f(zero)h(as)f Fi(i)c Fl(!)e(1)150 4250 y Fm(at)33 b(a)g(rate)g(that)g(is)g(indep)s (enden)m(t)i(of)f Fi(y)t Fm(.)349 4370 y(The)44 b(argumen)m(t)g(of)g (Step)g(2)f(sho)m(ws)i(that)e(the)h(maps)g Fi( )2515 4385 y Fj(n)2564 4395 y Fe(i)2644 4370 y Fm(are)f(uniformly)h(con)m (tin)m(uous)150 4489 y(with)37 b(estimates)g(that)g(are)g(indep)s (enden)m(t)i(of)f Fi(i)p Fm(.)55 b(As)38 b(a)e(result,)j(the)e(\014rst) g(term)h(can)f(b)s(e)h(made)150 4609 y(small,)d(indep)s(enden)m(t)i(of) f Fi(i)f Fm(\(and)g Fi(y)t Fm(\),)g(b)m(y)g(c)m(ho)s(osing)h Fi(r)h Fm(large)e(enough.)51 b(The)35 b(last)g(term)g(is)g(also)150 4729 y(small)k(if)h Fi(r)j Fm(is)c(large,)i(since)g Fi( )1309 4692 y Fh(0)1376 4729 y Fm(is)e(uniformly)h(con)m(tin)m(uous.)65 b(F)-8 b(or)39 b(\014xed)i Fi(r)h Fm(\(and)e(hence)h(\014xed)150 4857 y Fi(g)t Fm(\),)e(the)g(second)h(term)e(is)h(b)s(ounded)h(b)m(y)f Fi(C)7 b(r)s Fl(j)p Fm(1)26 b Fl(\000)g Fi(\025)2108 4813 y Fj(n)2157 4823 y Fe(i)2108 4884 y Fg(1)2194 4857 y Fi(\025)2252 4806 y Fh(\000)p Fj(k)r Fg(\()p Fj(n)2438 4816 y Fe(i)2470 4806 y Fg(\))2252 4884 y(2)2506 4857 y Fl(j)p Fm(,)40 b(whic)m(h)f(is)g(small)f(if)h Fi(i)g Fm(is)f(large)150 4977 y(enough.)44 b(Finally)-8 b(,)30 b(the)i(third)g(term)g(is)f(small)g(once)i Fi(i)f Fm(is)f(big)h(enough) g(that)f Fi( )3075 4992 y Fj(n)3124 5002 y Fe(i)3161 4977 y Fm(\()p Fi(x)p Fm(\))g(and)h Fi( )3588 4941 y Fh(0)3616 4977 y Fm(\()p Fi(x)p Fm(\))150 5097 y(agree)h(up)h(to)f(a)g (small)g(motion)f(on)i(a)f(ball)g(of)g(radius)h Fl(\035)27 b Fi(C)7 b(r)36 b Fm(ab)s(out)d(the)h(origin.)p 3266 5107 60 80 v 150 5280 a Fn(Prop)s(osition)k(1.)45 b Fm(F)-8 b(or)33 b(all)f(su\016cien)m(tly)i(large)e Fi(i)i Fm(w)m(e)g(ha)m(v)m (e)f Fi(\025)2464 5236 y Fj(n)2513 5246 y Fe(i)2464 5307 y Fg(1)2550 5280 y Fi(\025)2608 5228 y Fh(\000)p Fj(k)r Fg(\()p Fj(n)2794 5238 y Fe(i)2827 5228 y Fg(\))2608 5307 y(2)2890 5280 y Fm(=)c(1)p Fi(:)1925 5519 y Fm(9)p eop %%Page: 10 11 10 10 bop 150 100 a Fm(Pro)s(of.)42 b(Fix)28 b(a)h(tiling)e Fi(y)k Fl(2)d Fi(X)1225 115 y Fg(1)1298 100 y Fm(suc)m(h)j(that)d(for)h (some)g Fn(t)e Fl(2)h Fk(R)2361 63 y Fj(d)2421 100 y Fl(n)13 b(f)p Fn(O)p Fl(g)28 b Fm(and)i(for)f(some)g Fi(r)h(>)e(\032)13 b Fm(+)g Fi(S)3677 115 y Fg(0)3722 100 y Fi(;)150 219 y(y)43 b Fm(and)c Fi(y)30 b Fm(+)d Fn(t)39 b Fm(agree)g(on)g Fi(B)1194 234 y Fj(r)1238 219 y Fi(:)g Fm(Suc)m(h)i(a)e Fi(y)k Fm(exists)c(b)m(y)g(iii\).)61 b(Let)40 b Fi(\026)e Fl(2)g Fm([1)p Fi(;)17 b(\025)2938 234 y Fg(2)2981 219 y Fm(])39 b(b)s(e)h(a)f(limit)f(p)s(oin)m(t)150 348 y(of)31 b(\()p Fi(\025)358 312 y Fj(n)358 373 y Fg(1)412 348 y Fi(\025)470 296 y Fh(\000)p Fj(k)r Fg(\()p Fj(n)p Fg(\))470 375 y(2)693 348 y Fm(\))732 312 y Fh(1)732 373 y Fj(n)p Fg(=1)887 348 y Fi(:)f Fm(Then)i(there)g(is)e(an)h (increasing)h(sequence)g Fi(m)2650 363 y Fj(i)2715 348 y Fm(of)f(p)s(ositiv)m(e)f(in)m(tegers)h(suc)m(h)150 479 y(that)44 b Fi(\025)435 434 y Fj(m)506 444 y Fe(i)435 505 y Fg(1)542 479 y Fi(\025)600 427 y Fh(\000)p Fj(k)r Fg(\()p Fj(m)808 437 y Fe(i)840 427 y Fg(\))600 505 y(2)922 479 y Fl(!)h Fi(\026)f Fm(as)h Fi(i)h Fl(!)f(1)f Fm(and)h(suc)m(h)g (that)f Fi( )2406 494 y Fj(m)2477 504 y Fe(i)2513 479 y Fm(\()p Fi(y)t Fm(\))f(con)m(v)m(erges)i(to)f(a)g(tiling)f Fi(y)3695 443 y Fh(0)3722 479 y Fm(.)150 598 y(By)d(an)i(argumen)m(t)g (similar)e(to)h(Step)h(2)f(ab)s(o)m(v)m(e,)i Fi(y)2079 562 y Fh(0)2147 598 y Fm(and)f Fi(y)2402 562 y Fh(0)2457 598 y Fm(+)28 b Fi(\026)p Fn(t)41 b Fm(then)h(agree)f(on)h Fi(B)3434 616 y Fj(r)r(=)p Fg(2)3558 598 y Fi(:)f Fm(By)150 718 y(\014nite)28 b(lo)s(cal)e(complexit)m(y)-8 b(,)27 b(ho)m(w)m(ev)m(er,)i(there)f(are)g(only)f(\014nitely)g(man)m(y)g(v)-6 b(alues)28 b(of)f Fi(\026)h Fl(2)g Fm([1)p Fi(;)17 b(\025)3532 733 y Fg(2)3576 718 y Fm(])27 b(for)150 847 y(whic)m(h)36 b(this)e(is)h(p)s(ossible.)49 b(Th)m(us)36 b(the)f(image)f(of)h(the)g (sequence)i(\()p Fi(\025)2668 811 y Fj(n)2668 872 y Fg(1)2722 847 y Fi(\025)2780 795 y Fh(\000)p Fj(k)r Fg(\()p Fj(n)p Fg(\))2780 873 y(2)3003 847 y Fm(\))3042 811 y Fh(1)3042 872 y Fj(n)p Fg(=1)3231 847 y Fm(is)e(not)f(dense)150 966 y(in)d([1)p Fi(;)17 b(\025)445 981 y Fg(2)488 966 y Fm(])p Fi(:)31 b Fm(F)-8 b(rom)30 b(the)h(remarks)g(b)s(efore)h (Theorem)f(3)g(w)m(e)g(see)h(that)e(log)18 b Fi(\025)2897 981 y Fg(1)2941 966 y Fi(=)f Fm(log)g Fi(\025)3211 981 y Fg(2)3287 966 y Fm(is)30 b(rational,)150 1095 y(and)j(the)h(desired)g (result)f(follo)m(ws)g(from)g(the)g(fact)g(that)g(\()p Fi(\025)2372 1051 y Fj(n)2421 1061 y Fe(i)2372 1122 y Fg(1)2457 1095 y Fi(\025)2515 1044 y Fh(\000)p Fj(k)r Fg(\()p Fj(n)2701 1054 y Fe(i)2734 1044 y Fg(\))2515 1122 y(2)2770 1095 y Fm(\))2809 1059 y Fh(1)2809 1121 y Fj(i)p Fg(=1)2975 1095 y Fm(tak)m(es)g(only)f(\014nitely)150 1215 y(man)m(y)h(v)-6 b(alues)34 b(and)g(con)m(v)m(erges)g(to)f(1)p Fi(:)p 1650 1225 60 80 v 349 1334 a Fm(W)-8 b(e)26 b(ha)m(v)m(e)h(just) f(sho)m(wn)h(that)e Fi(\025)1486 1349 y Fg(1)1557 1334 y Fm(and)h Fi(\025)1801 1349 y Fg(2)1872 1334 y Fm(are)g(m)m (ultiplicativ)m(ely)e(dep)s(enden)m(t)k(o)m(v)m(er)e Fk(Q)p Fi(:)f Fm(Note)150 1458 y(that)32 b Fi( )430 1473 y Fj(n)479 1483 y Fe(i)548 1458 y Fm(is)g(a)g(conjugacy)g(whenev)m(er)i Fi(\025)1687 1414 y Fj(n)1736 1424 y Fe(i)1687 1484 y Fg(1)1800 1458 y Fm(=)29 b Fi(\025)1964 1406 y Fj(k)r Fg(\()p Fj(n)2088 1416 y Fe(i)2120 1406 y Fg(\))1964 1484 y(2)2156 1458 y Fi(:)j Fm(Denote)g(b)m(y)h Fl(O)s Fm([)p Fi(y)t Fm(])e(the)h(orbit)g(of)g(a)g(tiling)150 1577 y Fi(y)37 b Fm(under)d(translation.)150 1757 y Fn(Prop)s(osition)k (2.)45 b Fm(There)34 b(exists)f Fi(I)i Fl(2)28 b Fk(N)34 b Fm(suc)m(h)g(that)1493 2142 y Fi( )1558 2157 y Fj(n)1607 2167 y Fe(i)1643 2142 y Fm(\()p Fl(O)s Fm([)p Fi(y)t Fm(]\))26 b(=)j Fi( )2113 2101 y Fh(0)2140 2142 y Fm(\()p Fl(O)s Fm([)p Fi(y)t Fm(]\))1164 b(\(17\))150 2527 y(for)33 b(all)g Fi(i)28 b Fl(\025)g Fi(I)41 b Fm(and)34 b(for)f(all)f Fi(y)f Fl(2)e Fi(X)1435 2542 y Fg(1)1479 2527 y Fi(:)150 2707 y Fm(Pro)s(of.)42 b(If)31 b Fi(")f Fm(is)f(c)m(hosen)j(su\016cien) m(tly)e(small,)g(then)g(the)g(only)f(w)m(a)m(y)h(that)f(t)m(w)m(o)h (tilings)f Fi(y)t(;)17 b(y)3475 2671 y Fh(0)3529 2707 y Fl(2)28 b Fi(X)3706 2722 y Fg(2)150 2826 y Fm(can)40 b(ha)m(v)m(e)g(the)g(prop)s(ert)m(y)g(that)g Fi(d)p Fm(\()p Fi(g)t(y)t(;)17 b(g)t(y)1736 2790 y Fh(0)1759 2826 y Fm(\))39 b Fi(<)g(")h Fm(for)g(all)f Fi(g)i Fl(2)e(G)2596 2841 y Fj(E)2664 2826 y Fm(,)i(is)e(if)h Fi(y)j Fm(and)e Fi(y)3283 2790 y Fh(0)3350 2826 y Fm(are)f(small)150 2946 y(translates)23 b(of)h(one)g(another,)h(and)f(in)f(particular)g (that)g Fl(O)s Fm([)p Fi(y)t Fm(])k(=)h Fl(O)s Fm([)p Fi(y)2689 2910 y Fh(0)2716 2946 y Fm(])p Fi(:)23 b Fm(Cho)s(ose)g Fi(I)31 b Fm(large)23 b(enough)150 3075 y(so)39 b(that)f(for)h(all)f Fi(i)f Fl(\025)g Fi(I)8 b(;)38 b(\025)1162 3030 y Fj(n)1211 3040 y Fe(i)1162 3101 y Fg(1)1248 3075 y Fi(\025)1306 3023 y Fh(\000)p Fj(k)r Fg(\()p Fj(n)1492 3033 y Fe(i)1525 3023 y Fg(\))1306 3101 y(2)1598 3075 y Fm(=)f(1)h(and)i Fi(d)p Fm(\()p Fi( )2156 3090 y Fj(n)2205 3100 y Fe(i)2241 3075 y Fm(\()p Fi(y)t Fm(\))p Fi(;)17 b( )2486 3039 y Fh(0)2512 3075 y Fm(\()p Fi(y)t Fm(\)\))35 b Fi(<)i(")j Fm(for)e(all)g Fi(y)j Fl(2)c Fi(X)3494 3090 y Fg(1)3538 3075 y Fi(:)h Fm(F)-8 b(or)150 3194 y Fi(i)28 b Fl(\025)g Fi(I)41 b Fm(and)34 b Fi(y)d Fl(2)d Fi(X)853 3209 y Fg(1)930 3194 y Fm(w)m(e)34 b(ha)m(v)m(e)997 3579 y Fi(d)p Fm(\()p Fi(g)t( )1205 3594 y Fj(n)1254 3604 y Fe(i)1290 3579 y Fm(\()p Fi(y)t Fm(\))p Fi(;)17 b(g)t( )1587 3538 y Fh(0)1611 3579 y Fm(\()p Fi(y)t Fm(\)\))27 b(=)h Fi(d)p Fm(\()p Fi( )2069 3594 y Fj(n)2118 3604 y Fe(i)2154 3579 y Fm(\()p Fi(g)t(y)t Fm(\))p Fi(;)17 b( )2451 3538 y Fh(0)2476 3579 y Fm(\()p Fi(g)t(y)t Fm(\)\))25 b Fi(<)j(";)671 b Fm(\(18\))150 3964 y(th)m(us)34 b Fi( )434 3979 y Fj(n)483 3989 y Fe(i)520 3964 y Fm(\()p Fl(O)s Fm([)p Fi(y)t Fm(]\))26 b(=)i Fl(O)s Fm([)p Fi( )1095 3979 y Fj(n)1144 3989 y Fe(i)1180 3964 y Fm(\()p Fi(y)t Fm(\)])f(=)h Fl(O)s Fm([)p Fi( )1650 3928 y Fh(0)1677 3964 y Fm(\()p Fi(y)t Fm(\)])e(=)i Fi( )2036 3928 y Fh(0)2064 3964 y Fm(\()p Fl(O)s Fm([)p Fi(y)t Fm(]\))p Fi(:)p 2419 3974 V 150 4144 a Fn(Corollary)46 b(2.)64 b Fm(F)-8 b(or)39 b(eac)m(h)h Fi(i)f Fl(\025)g Fi(I)47 b Fm(and)40 b(eac)m(h)g Fi(y)i Fl(2)d Fi(X)2224 4159 y Fg(1)2268 4144 y Fm(,)i(there)f(exists)g Fi(g)2928 4159 y Fj(i;y)3066 4144 y Fm(in)g(the)g(cen)m(ter)h(of)150 4264 y Fl(G)209 4279 y Fj(r)r(el)317 4264 y Fm(\()p Fi(y)t Fm(\))i(suc)m(h)i(that)e Fi( )1017 4279 y Fj(n)1066 4289 y Fe(i)1103 4264 y Fm(\()p Fi(y)t Fm(\))h(=)i Fi(g)1449 4279 y Fj(i;y)1549 4264 y Fi( )1618 4227 y Fh(0)1645 4264 y Fm(\()p Fi(y)t Fm(\).)75 b(F)-8 b(urthermore,)47 b(if)d Fi(y)2653 4227 y Fh(0)2724 4264 y Fm(is)g(in)g(the)g(closure)h (of)f(the)150 4383 y(translational)32 b(orbit)h(of)g Fi(y)t Fm(,)g(then)g Fi(g)1477 4398 y Fj(i;y)1572 4378 y Fd(0)1631 4383 y Fm(=)c Fi(g)1785 4398 y Fj(i;y)1884 4383 y Fi(:)150 4563 y Fm(Pro)s(of.)91 b(Fix)49 b Fi(i)g Fm(and)h Fi(y)t(:)e Fm(By)g(Prop)s(osition)g(2)h(there)h(exists)f(a)g (translation)f Fi(g)3178 4578 y Fj(i;y)3327 4563 y Fm(suc)m(h)i(that) 150 4682 y Fi( )215 4697 y Fj(n)264 4707 y Fe(i)301 4682 y Fm(\()p Fi(y)t Fm(\))26 b(=)i Fi(g)611 4697 y Fj(i;y)711 4682 y Fi( )780 4646 y Fh(0)807 4682 y Fm(\()p Fi(y)t Fm(\))p Fi(:)j Fm(Since)j Fi( )1322 4697 y Fj(n)1371 4707 y Fe(i)1441 4682 y Fm(and)f Fi( )1703 4646 y Fh(0)1763 4682 y Fm(are)g(conjugacies)h(and)f(all)f(translations)h(comm)m(ute,) 150 4802 y(w)m(e)i(ha)m(v)m(e)g Fi(g)577 4817 y Fj(i;y)706 4802 y Fm(=)30 b Fi(g)861 4817 y Fj(i;y)956 4797 y Fd(0)1022 4802 y Fm(for)k(an)m(y)h Fi(y)1418 4766 y Fh(0)1480 4802 y Fm(in)f(the)h(translation)f(orbit)g(of)g Fi(y)t(:)g Fm(By)g(con)m(tin)m(uit)m(y)-8 b(,)34 b(this)g(last)150 4921 y(equalit)m(y)e(holds)i(for)f(all)g Fi(y)1134 4885 y Fh(0)1194 4921 y Fm(in)g(the)h(closure)g(of)f(the)h(translation)f (orbit)f(of)i Fi(y)t(:)349 5041 y Fm(T)-8 b(o)29 b(sho)m(w)h(that)f Fi(g)995 5056 y Fj(i;y)1123 5041 y Fm(is)h(in)f(the)h(cen)m(ter)g(of)g Fl(G)1961 5056 y Fj(r)r(el)2068 5041 y Fm(\()p Fi(y)t Fm(\))p Fi(;)e Fm(it)h(su\016ces)i(to)e(sho)m(w)h(that)f Fi(\013g)3371 5056 y Fj(i;y)3498 5041 y Fm(=)f Fi(g)3651 5056 y Fj(i;y)150 5160 y Fm(for)e(ev)m(ery)f Fi(\013)j Fl(2)g Fi(S)6 b(O)s Fm(\()p Fi(d)p Fm(\))g Fl(\\)g(G)1149 5175 y Fj(r)r(el)1257 5160 y Fm(\()p Fi(y)t Fm(\))p Fi(:)24 b Fm(Fix)h(suc)m(h)h Fi(\013:)g Fm(By)e(de\014nition)i(of)g Fl(G)2693 5175 y Fj(r)r(el)2800 5160 y Fm(\()p Fi(y)t Fm(\))f(there)h(is)f(a)g(sequence)150 5280 y Fi(h)207 5295 y Fj(j)281 5280 y Fl(2)33 b(G)439 5295 y Fj(r)r(el)546 5280 y Fm(\()p Fi(y)t Fm(\))i(suc)m(h)i(that)f Fi(h)1215 5295 y Fj(j)1257 5280 y Fi(y)f Fl(!)d Fi(y)39 b Fm(as)d Fi(j)i Fl(!)31 b(1)k Fm(and)i(suc)m(h)g(that)e(the)h(rotational)f(part) g Fi(\013)3592 5295 y Fj(j)3670 5280 y Fm(of)1900 5519 y(10)p eop %%Page: 11 12 11 11 bop 150 100 a Fi(h)207 115 y Fj(j)283 100 y Fm(con)m(v)m(erges)34 b(to)f Fi(\013:)g Fm(If)g Fi(g)1123 115 y Fj(i;y)1256 100 y Fm(is)g(translation)f(b)m(y)i Fn(s)2044 115 y Fj(i;y)2177 100 y Fm(then)1168 312 y Fi( )1237 271 y Fh(0)1264 312 y Fm(\()p Fi(y)t Fm(\))21 b(+)i Fn(s)1561 327 y Fj(i;y)1689 312 y Fm(=)28 b Fi( )1859 327 y Fj(n)1908 337 y Fe(i)1944 312 y Fm(\()p Fi(y)t Fm(\))1689 462 y(=)57 b(lim)1794 525 y Fj(j)t Fh(!1)2006 462 y Fi( )2071 477 y Fj(n)2120 487 y Fe(i)2157 462 y Fm(\()p Fi(h)2253 477 y Fj(j)2295 462 y Fi(y)t Fm(\))1689 663 y(=)g(lim)1794 725 y Fj(j)t Fh(!1)2006 663 y Fi(h)2063 678 y Fj(j)2106 663 y Fi( )2171 678 y Fj(n)2220 688 y Fe(i)2256 663 y Fm(\()p Fi(y)t Fm(\))1689 864 y(=)g(lim)1794 926 y Fj(j)t Fh(!1)2006 864 y Fi(h)2063 879 y Fj(j)2106 864 y Fi(g)2154 879 y Fj(i;y)2253 864 y Fi( )2322 822 y Fh(0)2349 864 y Fm(\()p Fi(y)t Fm(\))1689 1064 y(=)g(lim)1794 1127 y Fj(j)t Fh(!1)2006 1064 y Fi(h)2063 1079 y Fj(j)2106 1064 y Fi( )2175 1023 y Fh(0)2202 1064 y Fm(\()p Fi(y)t Fm(\))21 b(+)i Fi(\013)2518 1079 y Fj(j)2559 1064 y Fn(s)2604 1079 y Fj(i;y)1689 1265 y Fm(=)57 b(lim)1794 1328 y Fj(j)t Fh(!1)2006 1265 y Fi( )2075 1224 y Fh(0)2103 1265 y Fm(\()p Fi(h)2199 1280 y Fj(j)2241 1265 y Fi(y)t Fm(\))21 b(+)i Fi(\013)2518 1280 y Fj(j)2559 1265 y Fn(s)2604 1280 y Fj(i;y)2704 1265 y Fi(;)1689 1466 y Fm(=)28 b Fi( )1863 1425 y Fh(0)1890 1466 y Fm(\()p Fi(y)t Fm(\))21 b(+)52 b(lim)2142 1529 y Fj(j)t Fh(!1)2354 1466 y Fi(\013)2418 1481 y Fj(j)2460 1466 y Fn(s)2505 1481 y Fj(i;y)2605 1466 y Fi(;)3573 915 y Fm(\(19\))150 1722 y(and)34 b(hence)g Fi(\013)684 1737 y Fj(j)726 1722 y Fn(s)771 1737 y Fj(i;y)899 1722 y Fl(!)27 b Fn(s)1071 1737 y Fj(i;y)1171 1722 y Fi(:)33 b Fm(It)g(follo)m(ws)g(that)g Fi(g)1931 1737 y Fj(i;y)2063 1722 y Fm(comm)m(utes)h(with)f Fi(\013:)p 2943 1732 60 80 v 150 1901 a Fn(Corollary)39 b(3.)44 b Fm(The)34 b(original)e (conjugacy)i Fi( )i Fm(has)e(the)g(Little)e(Caesar)i(prop)s(ert)m(y)-8 b(.)150 2081 y(Pro)s(of.)71 b(First)43 b(w)m(e)g(sho)m(w)g(that)f Fi( )1444 2096 y Fj(n)1493 2106 y Fe(i)1572 2081 y Fm(has)h(the)g (Little)f(Caesar)g(prop)s(ert)m(y)h(for)g Fi(i)g Fl(\025)h Fi(I)8 b Fm(.)71 b(If)43 b(t)m(w)m(o)150 2200 y(tilings)36 b Fi(x;)17 b(y)36 b Fl(2)f Fi(X)824 2215 y Fg(1)905 2200 y Fm(agree)i(on)g(a)f(large)h(ball)f(around)i(the)f(origin,)g(they)g (lie)f(in)h(the)g(closure)h(of)150 2320 y(the)29 b(same)g (translational)e(orbit.)42 b(Since)30 b Fi( )1741 2335 y Fj(n)1790 2345 y Fb(1)1862 2320 y Fm(and)g Fi( )2121 2284 y Fh(0)2177 2320 y Fm(di\013er)f(b)m(y)f(a)h(\(\014xed\))f (translation)g(on)h(this)150 2439 y(orbit)h(closure,)h(and)f(since)h Fi( )1249 2403 y Fh(0)1277 2439 y Fm(\()p Fi(x)p Fm(\))e(and)i Fi( )1701 2403 y Fh(0)1728 2439 y Fm(\()p Fi(y)t Fm(\))e(agree)i(on)f (a)g(\(smaller\))f(ball)h(around)h(the)f(origin,)150 2559 y Fi( )215 2574 y Fj(n)264 2584 y Fe(i)301 2559 y Fm(\()p Fi(x)p Fm(\))i(and)i Fi( )727 2574 y Fj(n)776 2584 y Fe(i)813 2559 y Fm(\()p Fi(y)t Fm(\))e(agree)h(on)h(a)f(\(still) f(smaller\))g(ball)h(around)h(the)g(origin.)349 2688 y(No)m(w)25 b(note)h(that)f Fi( )31 b Fm(=)d Fi(\036)1250 2636 y Fj(k)r Fg(\()p Fj(n)1374 2646 y Fe(i)1406 2636 y Fg(\))1250 2715 y(2)1442 2688 y Fi( )1507 2703 y Fj(n)1556 2713 y Fe(i)1593 2688 y Fi(\036)1652 2644 y Fh(\000)p Fj(n)1763 2654 y Fe(i)1652 2715 y Fg(1)1826 2688 y Fm(is)d(a)g(comp)s (osition)g(of)g(three)h(maps,)h(eac)m(h)g(of)e(whic)m(h)150 2808 y(has)k(the)f(Little)g(Caesar)g(prop)s(ert)m(y)h(up)g(to)f (scaling.)42 b(Eac)m(h)28 b(patc)m(h)i(in)e Fi( )t Fm(\()p Fi(x)p Fm(\))f(is)h(determined)i(b)m(y)e(a)150 2927 y(\(m)m(uc)m(h)34 b(smaller\))e(patc)m(h)h(in)f Fi( )1289 2942 y Fj(n)1338 2952 y Fe(i)1396 2927 y Fl(\016)20 b Fi(\036)1525 2883 y Fh(\000)p Fj(n)1636 2893 y Fe(i)1525 2954 y Fg(1)1674 2927 y Fm(\()p Fi(x)p Fm(\),)31 b(whic)m(h)j(is)e(determined)i(b)m(y)e (a)h(patc)m(h)g(in)f Fi(\036)3439 2883 y Fh(\000)p Fj(n)3550 2893 y Fe(i)3439 2954 y Fg(1)3588 2927 y Fm(\()p Fi(x)p Fm(\),)150 3047 y(whic)m(h)37 b(is)g(determined)h(b)m(y)f(a)f(\(m)m(uc) m(h)i(larger\))e(patc)m(h)h(in)g Fi(x)p Fm(.)54 b(Th)m(us)38 b Fi( )i Fm(has)d(the)g(Little)f(Caesar)150 3166 y(prop)s(ert)m(y)-8 b(.)p 648 3176 V 150 3346 a Fn(Corollary)32 b(4.)43 b Fm(If)28 b(the)g(translational)f(orbit)g(of)h(a)f(tiling)g(is)h(dense)h (in)f Fi(X)2855 3361 y Fg(1)2899 3346 y Fm(,)h(then)f Fi( )j Fm(in)m(tert)m(wines)150 3465 y(the)j(actions)f(of)g(some)h(p)s (o)m(w)m(ers)g(of)f Fi(\036)1525 3480 y Fg(1)1604 3465 y Fm(and)g Fi(\036)1856 3480 y Fg(2)1901 3465 y Fi(:)150 3645 y Fm(Pro)s(of.)76 b(In)45 b(this)f(case)h Fl(G)1116 3660 y Fj(r)r(el)1223 3645 y Fm(\()p Fi(y)t Fm(\))f(of)g(a)g(tiling)f Fi(y)k Fm(has)e(no)f(cen)m(ter,)k(so)c Fi(g)2840 3660 y Fj(i)2917 3645 y Fm(is)g(the)h(iden)m(tit)m(y)e(and)150 3764 y Fi( )215 3779 y Fj(n)264 3789 y Fe(i)328 3764 y Fm(=)29 b Fi( )503 3728 y Fh(0)563 3764 y Fm(for)34 b(all)e Fi(i)c Fl(\025)g Fi(I)8 b Fm(.)44 b(But)33 b(then)628 3994 y Fi(\036)687 3942 y Fh(\000)p Fj(k)r Fg(\()p Fj(n)873 3952 y Fe(i)906 3942 y Fg(\))687 4020 y(2)964 3994 y Fl(\016)22 b Fi( )k Fl(\016)c Fi(\036)1258 3950 y Fj(n)1307 3960 y Fe(i)1258 4020 y Fg(1)1372 3994 y Fm(=)28 b Fi( )1542 4009 y Fj(n)1591 4019 y Fe(i)1655 3994 y Fm(=)g Fi( )1829 3953 y Fh(0)1884 3994 y Fm(=)h Fi( )2055 4009 y Fj(n)2104 4019 y Fe(i)p Fb(+1)2253 3994 y Fm(=)f Fi(\036)2417 3941 y Fh(\000)p Fj(k)r Fg(\()p Fj(n)2603 3951 y Fe(i)p Fb(+1)2721 3941 y Fg(\))2417 4020 y(2)2779 3994 y Fl(\016)22 b Fi( )k Fl(\016)c Fi(\036)3073 3941 y Fj(n)3122 3951 y Fe(i)p Fb(+1)3073 4020 y Fg(1)3244 3994 y Fi(:)301 b Fm(\(20\))150 4232 y(Multiplying)32 b(on)i(the)f(left)h(b)m(y)f Fi(\036)1374 4179 y Fj(k)r Fg(\()p Fj(n)1498 4189 y Fe(i)p Fb(+1)1616 4179 y Fg(\))1374 4258 y(2)1685 4232 y Fm(and)h(on)f(the)h(righ)m(t)f (b)m(y)g Fi(\036)2627 4187 y Fh(\000)p Fj(n)2738 4197 y Fe(i)2627 4258 y Fg(1)2809 4232 y Fm(giv)m(es)1163 4463 y Fi(\036)1222 4410 y Fj(k)r Fg(\()p Fj(n)1346 4420 y Fe(i)p Fb(+1)1463 4410 y Fg(\))p Fh(\000)p Fj(k)r Fg(\()p Fj(n)1680 4420 y Fe(i)1713 4410 y Fg(\))1222 4489 y(2)1771 4463 y Fl(\016)22 b Fi( )31 b Fm(=)d Fi( )e Fl(\016)c Fi(\036)2266 4410 y Fj(n)2315 4420 y Fe(i)p Fb(+1)2432 4410 y Fh(\000)p Fj(n)2543 4420 y Fe(i)2266 4489 y Fg(1)2580 4463 y Fi(:)p 2667 4473 V 965 w Fm(\(21\))349 4802 y(W)-8 b(e)34 b(no)m(w)g(pro)s(ceed)h(to)e(the)h(pro)s(of)g(of)g(Theorem)h(2.) 45 b(W)-8 b(e)34 b(abbreviate)f Fi(X)3057 4817 y Fj(\036)3144 4802 y Fm(b)m(y)g Fi(X)41 b Fm(and)35 b(call)150 4921 y(a)30 b(patc)m(h)g(admissable)g(if)g(there)h(is)e(a)h(tiling)f Fi(x)e Fl(2)h Fi(X)38 b Fm(con)m(taining)29 b(it.)43 b(W)-8 b(e)30 b(assume)h Fi(\036)f Fm(has)g(a)f(\014xed)150 5041 y(p)s(oin)m(t)i(in)h Fi(X)8 b Fm(,)31 b(not)g(\014xed)h(b)m(y)g (an)m(y)f(translation,)g(whose)h(orbit)g(under)g(translations)f(is)h (dense)h(in)150 5160 y Fi(X)8 b Fm(.)60 b(\(The)39 b(existence)g(of)g (a)g(\014xed)g(p)s(oin)m(t)g(for)g Fi(\036)g Fm(follo)m(ws)g(from)g (iii\).\))59 b(Let)39 b Fi(H)45 b Fm(:)36 b Fl(G)3241 5175 y Fj(E)3346 5160 y Fl(7!)g(G)3541 5175 y Fj(E)3648 5160 y Fm(b)s(e)150 5280 y(de\014ned)f(b)m(y)f Fi(\036)p Fm(\()p Fi(g)t(x)p Fm(\))26 b(=)i Fi(H)8 b Fm(\()p Fi(g)t Fm(\))p Fi(\036)p Fm(\()p Fi(x)p Fm(\))p Fi(:)1900 5519 y Fm(11)p eop %%Page: 12 13 12 12 bop 349 100 a Fm(The)41 b(follo)m(wing)e(four)i(lemmas)f(are)g (pro)m(v)m(ed)h(with)f(standard)h(argumen)m(ts,)h(as)e(sk)m(etc)m(hed) 150 219 y(b)s(elo)m(w.)150 399 y Fn(Lemma)e(3.)45 b Fm(The)33 b(extension)h Fi(\036)28 b Fm(:)f Fi(X)35 b Fl(7!)28 b Fi(X)40 b Fm(is)33 b(surjectiv)m(e.)150 578 y(Sk)m(etc)m(h)e(of)g (pro)s(of.)43 b(Since)31 b Fi(X)37 b Fm(is)31 b(compact,)f Fi(\036)p Fm(\()p Fi(X)8 b Fm(\))30 b(is)g(a)g(closed)h(subset)g(of)g Fi(X)8 b Fm(.)42 b(T)-8 b(o)30 b(see)h(that)e(it)h(is)150 698 y(dense,)36 b(note)f(that)f(an)m(y)g(admissible)h(patc)m(h)g(is)g (a)f(subset)i(of)f(some)g(tile)f(of)g(lev)m(el)g Fi(n)p Fm(.)48 b(Th)m(us,)36 b(for)150 821 y(an)m(y)d(tiling)f Fi(x)c Fl(2)g Fi(X)8 b Fm(,)32 b(and)i(an)m(y)f Fi(r)e(>)d Fm(0,)k Fi(B)1682 785 y Fj(x)1677 846 y(r)1760 821 y Fm(=)c Fi(B)1946 770 y Fj(\036)p Fg(\()p Fj(y)s Fg(\))1941 833 y Fj(r)2137 821 y Fm(for)33 b(some)h(tiling)e Fi(y)t Fm(.)p 2983 831 60 80 v 150 1001 a Fn(Lemma)38 b(4.)44 b Fm(There)33 b(is)g(a)f(constan)m(t)i Fi(C)h(>)28 b Fm(0)k(suc)m(h)j(that)d(for)h(ev)m(ery)f Fi(r)f(>)d Fm(0)k(and)i(for)f (ev)m(ery)f(pair)150 1120 y(of)i(admissable)g(patc)m(hes)h Fi(P)s(;)17 b(P)1317 1084 y Fh(0)1378 1120 y Fm(with)66 b(supp)r(\()p Fi(P)14 b Fm(\))27 b Fl(\032)i Fi(B)2209 1135 y Fj(r)2287 1120 y Fm(and)34 b Fi(B)2557 1135 y Fj(C)5 b(r)2691 1120 y Fl(\032)62 b Fm(supp)r(\()p Fi(P)3153 1084 y Fh(0)3180 1120 y Fm(\))34 b(there)g(exists)150 1240 y Fi(g)d Fl(2)d(G)382 1255 y Fj(E)483 1240 y Fm(suc)m(h)34 b(that)f Fi(g)t(P)41 b Fl(\032)28 b Fi(P)1263 1204 y Fh(0)1291 1240 y Fi(:)150 1419 y Fm(Sk)m(etc)m(h)33 b(of)f(pro)s(of.)44 b(By)31 b(minimalit)m(y)-8 b(,)30 b(this)i(is)f(true)i(for)f(patc)m (hes)h Fi(P)46 b Fm(of)32 b(size)g(less)g(than)h(10)e(times)150 1539 y(the)39 b(diameter)g(of)g(the)g(largest)g(tile.)60 b(By)38 b(Lemma)h(3,)h(one)f(can)g(desubstitute)h(\(p)s(erhaps)g(not) 150 1659 y(uniquely\),)31 b(thereb)m(y)i(relating)d(the)i(frequency)h (of)f(large)f(patc)m(hes)i(to)e(the)h(frequency)h(of)f(small)150 1778 y(patc)m(hes.)p 562 1788 V 150 1958 a Fn(Lemma)48 b(5.)69 b Fm(If)42 b Fn(b)896 1973 y Fg(1)940 1958 y Fi(;)17 b(:)g(:)g(:)d(;)j Fn(b)1226 1973 y Fj(k)1316 1958 y Fl(2)42 b Fk(E)1490 1922 y Fj(d)1578 1958 y Fm(and)g Fn(t)f Fm(=)1985 1883 y Fa(P)2090 1908 y Fj(k)2090 1988 y(i)p Fg(=1)2241 1958 y Fi(a)2294 1973 y Fj(i)2327 1958 y Fn(b)2391 1973 y Fj(i)2465 1958 y Fm(with)g Fi(a)2753 1973 y Fj(i)2828 1958 y Fl(2)h Fk(N)f Fm(then)h(there)g(exist)150 2077 y Fn(t)195 2092 y Fg(0)239 2077 y Fi(;)17 b Fn(t)329 2092 y Fg(1)373 2077 y Fi(;)g(:)g(:)g(:)d(;)j Fn(t)640 2092 y Fj(`)710 2077 y Fm(suc)m(h)35 b(that)e Fn(t)1196 2092 y Fg(0)1268 2077 y Fm(=)28 b(0)p Fi(;)33 b Fn(t)1529 2092 y Fj(`)1594 2077 y Fm(=)28 b Fn(t)p Fi(;)33 b Fm(and)g(for)h(eac)m (h)g Fi(j)g Fm(=)28 b(1)p Fi(;)17 b(:)g(:)g(:)d(;)j(`;)1266 2301 y Fn(t)1311 2316 y Fj(j)1374 2301 y Fl(\000)23 b Fn(t)1519 2316 y Fj(j)t Fh(\000)p Fg(1)1690 2301 y Fl(2)28 b(f\006)p Fn(b)1975 2316 y Fj(i)2036 2301 y Fm(:)f Fi(i)h Fm(=)g(1)p Fi(;)17 b(:)g(:)g(:)d(;)j(k)s Fl(g)938 b Fm(\(22\))150 2524 y(and)34 b Fn(t)389 2539 y Fj(j)463 2524 y Fm(lies)g(within)f(\()p Fi(k)s(=)p Fm(2\))17 b(max)1380 2539 y Fg(1)p Fh(\024)p Fj(i)p Fh(\024)p Fj(k)1638 2524 y Fl(k)p Fn(b)1752 2539 y Fj(i)1784 2524 y Fl(k)33 b Fm(of)h(the)f(straigh)m(t-line)h(path)f (from)g Fn(O)h Fm(to)f Fn(t)p Fi(:)150 2703 y Fm(Sk)m(etc)m(h)43 b(of)g(pro)s(of.)71 b(Eac)m(h)43 b(p)s(oin)m(t)f(along)f(the)i(straigh) m(t-line)f(path)h(from)f Fn(O)g Fm(to)g Fn(t)g Fm(is)g(a)g(linear)150 2823 y(com)m(bination)25 b(of)f(the)h Fn(b)1040 2838 y Fj(i)1073 2823 y Fm('s)f(with)h(real)f(co)s(e\016cien)m(ts.)42 b(Round)25 b(these)h(co)s(e\016cien)m(ts)f(to)f(the)h(nearest)150 2943 y(in)m(teger)34 b(to)e(get)h(the)h(sequence)h(of)e Fn(t)1510 2958 y Fj(j)1552 2943 y Fm('s.)p 1706 2953 V 150 3122 a Fn(Lemma)49 b(6.)72 b Fm(If)42 b Fl(G)895 3137 y Fj(r)r(el)1003 3122 y Fm(\()p Fi(y)t Fm(\))g(=)h Fi(S)6 b(O)s Fm(\()p Fi(d)p Fm(\))42 b(there)h(is)f(a)g(constan)m(t)h Fi(D)j Fm(suc)m(h)e(that)e(if)g Fi(P)s(;)17 b(P)3367 3086 y Fh(0)3436 3122 y Fm(are)43 b(ad-)150 3242 y(missable)c(patc)m (hes)g(in)f Fi(y)k Fm(with)71 b(supp)q(\()p Fi(P)14 b Fm(\))35 b Fl(\032)h Fi(B)1979 3257 y Fj(r)2061 3242 y Fm(and)j Fi(B)2336 3257 y Fj(D)r(r)2483 3242 y Fl(\032)69 b Fm(supp)q(\()p Fi(P)2951 3206 y Fh(0)2979 3242 y Fm(\))38 b(then)h(there)f(exist)150 3361 y Fi(\013)214 3376 y Fg(1)258 3361 y Fi(;)17 b(:)g(:)g(:)e(;)i(\013)545 3376 y Fj(n)634 3361 y Fl(2)38 b Fi(S)6 b(O)s Fm(\()p Fi(d)p Fm(\))37 b(and)i Fn(t)1295 3376 y Fg(1)1339 3361 y Fi(;)17 b(:)g(:)g(:)e(;)i Fn(t)1607 3376 y Fj(n)1696 3361 y Fl(2)37 b Fk(R)1871 3325 y Fj(d)1956 3361 y Fm(suc)m(h)k(that)d Fi(\013)2472 3376 y Fj(i)2505 3361 y Fi(P)h Fm(+)26 b Fn(t)2756 3376 y Fj(i)2826 3361 y Fl(\032)37 b Fi(P)3018 3325 y Fh(0)3046 3361 y Fi(;)h(i)f Fm(=)g(1)p Fi(;)17 b Fm(2)p Fi(;)g(:)g(:)g(:)c(;)k(n;)150 3481 y Fm(and)34 b(suc)m(h)g(that)f(no)h(prop)s(er)f(subspace)j(of)d Fk(E)1837 3445 y Fj(d)1917 3481 y Fm(is)g(in)m(v)-6 b(arian)m(t)34 b(under)g(all)e(the)i Fi(\013)3092 3496 y Fj(i)3125 3481 y Fi(:)150 3660 y Fm(Sk)m(etc)m(h)j(of)e(pro)s(of.)51 b(This)36 b(is)g(similar)e(to)h(Lemma)h(4.)51 b(W)-8 b(e)36 b(pro)m(v)m(e)g(it)f(\014rst)h(for)g(individual)f(tiles,)150 3780 y(and)f(then)g(extend)f(the)h(result)g(to)e(larger)h(patc)m(hes)i (b)m(y)e(desubstitution.)p 2953 3790 V 349 4019 a(Let)f Fi(M)38 b Fm(=)28 b(max)p Fl(f)p Fm(diam)o(\()p Fi(T)14 b Fm(\))28 b(:)f Fi(T)42 b Fl(2)28 b(Ag)p Fi(:)i Fm(Let)i(us)g(sa)m(y)f (that)h(a)f(patc)m(h)h Fi(P)45 b Fm(has)32 b Fc(p)-5 b(erio)g(d)31 b Fi(g)f Fl(2)e(G)3682 4034 y Fj(E)150 4139 y Fm(if)39 b Fi(P)g Fl([)26 b Fi(g)t(P)52 b Fm(is)38 b(a)h(patc)m(h,)h(i.e.,)f(if)f Fi(P)53 b Fm(and)39 b Fi(g)t(P)51 b Fm(agree)39 b(where)g(their)g(supp)s(orts)h(o)m(v)m (erlap)e(\(w)m(e)h(do)150 4259 y(not)c(require)h(that)f(they)g (actually)g(o)m(v)m(erlap\).)50 b(Alternativ)m(ely)-8 b(,)33 b Fi(P)50 b Fm(has)35 b(p)s(erio)s(d)h Fi(g)e Fl(2)e(G)3393 4274 y Fj(E)3496 4259 y Fm(if)j(and)150 4378 y(only)f(if)g(whenev)m(er)i Fi(T)43 b Fl(2)30 b Fi(P)48 b Fm(is)34 b(suc)m(h)i(that)e Fi(g)t(T)1878 4342 y Fh(\016)1957 4378 y Fm(in)m(tersects)69 b(supp)q(\()p Fi(P)14 b Fm(\))34 b(w)m(e)h(also)e(ha)m(v)m(e)i Fi(g)t(T)43 b Fl(2)30 b Fi(P)s(:)150 4498 y Fm(Of)k(course)g(an)m(y)g(subpatc)m(h)h (of)e(a)g(patc)m(h)h(of)f(p)s(erio)s(d)h Fi(g)i Fm(has)e(p)s(erio)s(d)f Fi(g)t(:)150 4677 y Fn(Lemma)f(7.)43 b Fm(If)28 b Fl(f)p Fn(b)890 4692 y Fg(1)934 4677 y Fi(;)17 b(:)g(:)g(:)d(;)j Fn(b)1220 4692 y Fj(k)r Fh(\000)p Fg(1)1370 4677 y Fl(g)28 b(\032)g Fk(E)1619 4641 y Fj(d)1694 4677 y Fm(is)g(a)f(basis)i(for)f(a) g(lattice)f Fl(L)h Fm(and)g Fi(P)42 b Fm(is)28 b(a)f(patc)m(h)i(ha)m (ving)150 4797 y(all)40 b(p)s(erio)s(ds)h(in)f Fl(L)g Fm(and)h(additional)f(translational)f(p)s(erio)s(d)i Fn(b)2535 4812 y Fj(k)2624 4797 y Fm(suc)m(h)h(that)e Fi(B)3155 4812 y Fj(r)3239 4797 y Fl(\032)73 b Fm(supp)r(\()p Fi(P)14 b Fm(\))150 4916 y(where)1337 5036 y Fi(r)30 b(>)e Fm(\()p Fi(d)22 b Fm(+)h(1\))41 b(max)1836 5101 y Fg(1)p Fh(\024)p Fj(i)p Fh(\024)p Fj(k)2088 5036 y Fl(k)p Fn(b)2202 5051 y Fj(i)2234 5036 y Fl(k)22 b Fm(+)h(4)p Fi(M)1020 b Fm(\(23\))150 5270 y(then)34 b Fl(h)p Fn(b)480 5285 y Fg(1)524 5270 y Fi(;)17 b(:)g(:)g(:)d(;)j Fn(b)810 5285 y Fj(k)858 5270 y Fl(i)33 b Fm(is)g(a)g(lattice)g(of)g(p)s(erio)s (ds)h(for)f Fi(B)2114 5234 y Fj(P)2109 5302 y(r)r(=)p Fg(2)2233 5270 y Fi(:)1900 5519 y Fm(12)p eop %%Page: 13 14 13 13 bop 150 100 a Fm(Pro)s(of.)42 b(Supp)s(ose)31 b Fi(T)42 b Fl(2)28 b Fi(B)1122 63 y Fj(P)1117 132 y(r)r(=)p Fg(2)1270 100 y Fm(and)i Fi(T)1532 63 y Fh(\016)1592 100 y Fm(+)14 b Fn(t)30 b Fm(in)m(tersects)64 b(supp)q(\()p Fi(B)2554 63 y Fj(P)2549 132 y(r)r(=)p Fg(2)2673 100 y Fm(\))p Fi(;)29 b Fm(where)h Fn(t)d Fm(=)h Fi(a)3283 115 y Fg(1)3328 100 y Fn(b)3392 115 y Fg(1)3450 100 y Fm(+)14 b Fi(:)j(:)g(:)d Fm(+)150 219 y Fi(a)203 234 y Fj(k)252 219 y Fn(b)316 234 y Fj(k)403 219 y Fm(with)39 b(eac)m(h)h Fi(a)919 234 y Fj(i)989 219 y Fl(2)d Fk(N)p Fi(:)i Fm(Let)g Fn(b)e Fl(2)h Fi(T)i Fl(\\)26 b Fi(B)1886 237 y Fj(r)r(=)p Fg(2)2010 219 y Fi(:)39 b Fm(Then)h Fn(b)25 b Fm(+)i Fn(t)37 b Fl(2)g Fi(B)2797 237 y Fj(r)r(=)p Fg(2+2)p Fj(M)3106 219 y Fi(:)h Fm(Let)h Fn(t)3401 234 y Fg(0)3446 219 y Fi(;)17 b(:)g(:)g(:)d(;)j Fn(t)3713 234 y Fj(`)150 339 y Fm(b)s(e)35 b(as)f(in)g(Lemma)g(5.)47 b(The)35 b(straigh)m(t-line)f(path)g(from)g Fn(b)g Fm(to)g Fn(b)22 b Fm(+)h Fn(t)34 b Fm(lies)g(in)g Fi(B)3093 357 y Fj(r)r(=)p Fg(2+2)p Fj(M)3436 339 y Fm(so)g(eac)m(h)150 458 y Fn(b)20 b Fm(+)h Fn(t)377 473 y Fj(j)451 458 y Fm(is)33 b(in)f Fi(B)742 473 y Fj(r)786 458 y Fi(:)g Fm(Th)m(us,)i(for)e(eac)m(h)i Fi(j)k Fm(w)m(e)33 b(ha)m(v)m(e)g(\()p Fi(T)2066 422 y Fh(\016)2132 458 y Fm(+)21 b Fn(t)2275 473 y Fj(j)2317 458 y Fm(\))f Fl(\\)h Fi(P)42 b Fl(6)p Fm(=)28 b Fl(;)k Fm(and)h(b)m(y)g(\014nite)f(induction)150 578 y Fi(T)k Fm(+)23 b Fn(t)389 593 y Fj(j)458 578 y Fl(2)28 b Fi(P)14 b Fm(.)44 b(It)33 b(follo)m(ws)g(that)g Fi(T)j Fm(+)23 b Fn(t)k Fl(2)h Fi(B)1794 542 y Fj(P)1789 610 y(r)r(=)p Fg(2)1913 578 y Fi(:)p 2001 588 60 80 v 150 762 a Fn(Lemma)f(8.)41 b Fm(If)23 b Fi(k)31 b(<)d(d)23 b Fm(and)h Fl(f)p Fn(b)1325 777 y Fg(1)1369 762 y Fi(;)17 b(:)g(:)g(:)d(;)j Fn(b)1655 777 y Fj(k)1703 762 y Fl(g)23 b Fm(is)g(a)g(basis)h(for)f(a)g(lattice)f Fl(L)h Fm(and)h Fi(P)37 b Fm(is)23 b(an)g(admissable)150 882 y(patc)m(h)34 b(ha)m(ving)f(all)f(p)s(erio)s(ds)i(in)f Fl(L)g Fm(suc)m(h)i(that)e Fi(B)1968 897 y Fj(r)2039 882 y Fl(\032)61 b Fm(supp)r(\()p Fi(P)14 b Fm(\))32 b(where)1255 1100 y Fi(r)f(>)d(D)s Fm(\(\()p Fi(d)22 b Fm(+)g(1\))42 b(max)1878 1165 y Fg(1)p Fh(\024)p Fj(i)p Fh(\024)p Fj(k)2130 1100 y Fl(k)p Fn(b)2244 1115 y Fj(i)2277 1100 y Fl(k)22 b Fm(+)h(4)p Fi(M)11 b Fm(\))927 b(\(24\))150 1371 y(then)29 b(there)g(exists)f Fn(b)956 1386 y Fj(k)r Fg(+1)1133 1371 y Fl(2)h Fk(E)1294 1335 y Fj(d)1353 1371 y Fl(n)45 b Fm(span)q(\()p Fl(L)p Fm(\))27 b(with)h Fl(k)p Fn(b)2158 1386 y Fj(k)r Fg(+1)2308 1371 y Fl(k)f(\024)h Fm(max)2676 1386 y Fg(1)p Fh(\024)p Fj(i)p Fh(\024)p Fj(k)2934 1371 y Fl(k)p Fn(b)3048 1386 y Fj(i)3081 1371 y Fl(k)g Fm(and)h(suc)m(h)g(that)150 1491 y Fl(h)p Fn(b)253 1506 y Fg(1)297 1491 y Fi(;)17 b(:)g(:)g(:)d(;)j Fn(b)583 1506 y Fj(k)r Fg(+1)732 1491 y Fl(i)33 b Fm(is)g(a)g(lattice)g(of)g(p)s(erio)s(ds)h(for)f Fi(B)1988 1455 y Fj(P)1983 1523 y(r)r(=)p Fg(3)p Fj(D)2175 1491 y Fi(:)150 1685 y Fm(Pro)s(of.)60 b(Let)39 b Fi(P)740 1649 y Fh(0)804 1685 y Fm(=)f Fi(B)1000 1649 y Fj(P)995 1717 y(r)r(=D)r Fh(\000)p Fj(M)1292 1685 y Fi(:)g Fm(By)g(Lemma)h(6,)h (there)f(exist)f Fi(\013)f Fl(2)g Fi(S)6 b(O)s Fm(\()p Fi(d)p Fm(\))38 b(and)h Fn(t)d Fl(2)h Fk(R)3473 1649 y Fj(d)3558 1685 y Fm(suc)m(h)150 1809 y(that)30 b Fi(\013P)505 1773 y Fh(0)549 1809 y Fm(+)16 b Fn(t)28 b Fl(\032)g Fi(P)45 b Fm(and)30 b Fi(\013)k Fm(span\()p Fl(L)p Fm(\))27 b Fl(6)p Fm(=)62 b(span)q(\()p Fl(L)p Fm(\))p Fi(:)29 b Fm(It)h(follo)m(ws)g(that)g Fi(P)2853 1773 y Fh(0)2911 1809 y Fm(has)h(all)e(p)s(erio)s(ds)i(in)f Fl(L)150 1929 y Fm(as)k(w)m(ell)g(as)f(all)g(p)s(erio)s(ds)i(in)e Fi(\013)1268 1893 y Fh(\000)p Fg(1)1375 1929 y Fl(L)p Fi(:)h Fm(Let)g Fn(b)1749 1944 y Fj(k)r Fg(+1)1927 1929 y Fl(2)29 b Fi(\013)2086 1893 y Fh(\000)p Fg(1)2193 1929 y Fl(f)p Fn(b)2307 1944 y Fg(1)2351 1929 y Fi(;)17 b(:)g(:)g(:)d(;)j Fn(b)2637 1944 y Fj(k)2685 1929 y Fl(g)23 b(n)55 b Fm(span)q(\()p Fl(L)p Fm(\))p Fi(:)33 b Fm(By)g(Lemma)150 2048 y(7,)e Fl(h)p Fn(b)362 2063 y Fg(1)406 2048 y Fi(;)17 b(:)g(:)g(:)d(;)j Fn(b)692 2063 y Fj(k)r Fg(+1)841 2048 y Fl(i)31 b Fm(is)g(a)g(lattice)g (of)g(p)s(erio)s(ds)h(for)f Fi(B)2083 2012 y Fj(P)2078 2080 y(r)r(=)p Fg(2)p Fj(D)r Fh(\000)p Fj(M)s(=)p Fg(2)2492 2048 y Fi(:)g Fm(Since)h Fi(r)s(=)p Fm(2)p Fi(D)21 b Fl(\000)d Fi(M)5 b(=)p Fm(2)28 b Fi(>)g(r)s(=)p Fm(3)p Fi(D)r(;)150 2168 y Fm(the)34 b(pro)s(of)f(is)g(complete.)p 1157 2178 V 150 2348 a Fn(Prop)s(osition)41 b(3.)50 b Fm(There)36 b(is)g(a)f(constan)m(t)g Fi(K)j(>)32 b Fm(0)j(suc)m(h)i (that)d(if)i Fi(P)49 b Fm(is)35 b(an)g(admissable)h(patc)m(h)150 2467 y(whose)44 b(supp)s(ort)f(con)m(tains)h(a)f(ball)g(of)g(radius)g Fi(r)j Fm(then)e(ev)m(ery)f(non-iden)m(tit)m(y)h(p)s(erio)s(d)f Fi(g)j Fm(of)d Fi(P)150 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Fm(for)34 b(all)e Fn(b)c Fl(2)61 b Fm(supp)q(\()p Fi(B)2966 5221 y Fj(x)2961 5296 y(r)r Fg(\()3052 5269 y Fb(1)p 3043 5280 V 3043 5320 a Fe(C)3108 5296 y Fh(\000)p Fj(K)5 b Fg(\))p Fh(\000)p Fj(M)3424 5262 y Fm(\))p Fi(:)82 b Fm(\(27\))1900 5519 y(13)p eop %%Page: 14 15 14 14 bop 150 100 a Fm(No)m(w)44 b Fi(H)8 b Fm(\()p Fi(h)575 63 y Fh(\000)p Fg(1)682 100 y Fi(g)t(h)p Fm(\)\()p Fi(h)926 63 y Fh(\000)p Fg(1)1033 100 y Fi(g)t(h)p Fm(\))1181 63 y Fh(\000)p Fg(1)1331 100 y Fm(is)45 b(a)f(translation,)j(sa)m(y)d (b)m(y)h Fn(b)2481 115 y Fg(1)2572 100 y Fl(2)i Fk(R)2757 63 y Fj(d)2803 100 y Fi(;)d Fm(and)h(if)g Fn(b)3247 115 y Fg(1)3338 100 y Fm(=)i(0)d(then)150 219 y Fi(h)207 183 y Fh(\000)p Fg(1)314 219 y Fi(g)t(h)c Fl(2)g Fi(S)6 b(O)s Fm(\()p Fi(d)p Fm(\))40 b(and)h Fi(h)1143 183 y Fh(\000)p Fg(1)1251 219 y Fi(g)t(hb)e Fm(=)h Fi(b:)g Fm(This)h(last)g(is)f(imp)s(ossible)h(due)h(to)e(our)h(c)m(hoice)g(of)g Fi(K)r(;)150 339 y Fm(hence)1457 469 y(0)28 b Fi(<)g Fl(k)p Fi(x)1747 484 y Fg(1)1791 469 y Fl(k)g(\024)g Fm(\()p Fi(\025)22 b Fm(+)h(1\))p Fi(K)7 b(r)m(:)1129 b Fm(\(28\))349 670 y(An)43 b(application)g(of)h(Lemma)f(7)g(follo)m(w) m(ed)h(b)m(y)f Fi(d)29 b Fl(\000)g Fm(1)43 b(applications)g(of)g(Lemma) h(8)f(\(w)m(e)150 789 y(will)34 b(see)i(in)f(a)g(momen)m(t)h(that)f Fi(r)j Fm(is)d(large)g(enough)h(for)f(this\))g(yields)g(a)g Fi(d)p Fm(-dimensional)h(lattice)150 909 y Fl(L)27 b Fm(=)i Fl(h)p Fn(b)455 924 y Fg(1)499 909 y Fi(;)17 b(:)g(:)g(:)d(;)j Fn(b)785 924 y Fj(d)830 909 y Fl(i)33 b Fm(of)h(p)s(erio)s(ds)f(for)h Fi(P)1596 873 y Fh(0)1651 909 y Fm(=)28 b Fi(B)1837 873 y Fj(x)1832 936 y(r)1871 916 y Fd(0)1903 909 y Fm(,)33 b(where)1184 1159 y Fl(k)p Fn(b)1298 1174 y Fj(i)1331 1159 y Fl(k)27 b(\024)h Fm(\()p Fi(\025)22 b Fm(+)h(1\))p Fi(K)7 b(r)m(;)215 b(i)28 b Fm(=)g(1)p Fi(;)17 b(:)g(:)g(:)d(;)j(d;)856 b Fm(\(29\))150 1409 y(and)1293 1552 y Fi(r)1341 1510 y Fh(0)1397 1552 y Fm(=)1694 1484 y(1)p 1514 1529 411 4 v 1514 1620 a(2\(3)p Fi(D)s Fm(\))1777 1591 y Fj(d)p Fh(\000)p Fg(1)1936 1552 y Fi(r)s Fm(\()2049 1484 y(1)p 2035 1529 79 4 v 2035 1620 a Fi(C)2147 1552 y Fl(\000)23 b Fi(K)7 b Fm(\))22 b Fl(\000)g Fi(M)1397 1785 y Fl(\025)1734 1718 y Fi(r)p 1514 1762 490 4 v 1514 1853 a Fm(4)p Fi(C)7 b Fm(\(3)p Fi(D)s Fm(\))1855 1825 y Fj(d)p Fh(\000)p Fg(1)1397 1993 y Fl(\025)28 b Fm(\()p Fi(d)22 b Fm(+)g(4)p Fi(M)5 b(=m)p Fm(\))p Fi(\025)p Fm(\()p Fi(\025)22 b Fm(+)h(1\))p Fi(K)7 b(r)1397 2142 y Fl(\025)28 b Fm(\()p Fi(d)22 b Fm(+)g(4)p Fi(M)5 b(=m)p Fm(\))p Fi(\025)41 b Fm(max)2117 2207 y Fg(1)p Fh(\024)p Fj(i)p Fh(\024)p Fj(d)2367 2142 y Fl(k)p Fi(x)2474 2157 y Fj(i)2507 2142 y Fl(k)p Fi(:)3573 1850 y Fm(\(30\))150 2409 y(Th)m(us)34 b Fi(r)450 2373 y Fh(0)505 2409 y Fi(>)28 b(\025)p Fm(\()707 2335 y Fa(P)812 2359 y Fj(d)812 2439 y(i)p Fg(=1)963 2409 y Fl(k)p Fi(x)1070 2424 y Fj(i)1103 2409 y Fl(k)20 b Fm(+)h Fi(M)11 b Fm(\))p Fi(;)32 b Fm(so)g(the)h(fundamen)m(tal)h (domain)f Fi(F)41 b Fm(=)28 b Fl(f)p Fi(t)2995 2424 y Fg(1)3040 2409 y Fn(b)3104 2424 y Fg(1)3169 2409 y Fm(+)21 b Fi(:)c(:)g(:)i Fm(+)i Fi(t)3538 2424 y Fj(d)3585 2409 y Fn(b)3649 2424 y Fj(d)3722 2409 y Fm(:)150 2529 y Fi(t)186 2544 y Fj(i)248 2529 y Fl(2)29 b Fm([0)p Fi(;)17 b Fm(1])g Fl(g)31 b Fm(for)j Fl(L)g Fm(is)g(suc)m(h)h(that)e Fi(\036)p Fm(\()p Fi(F)1616 2493 y Fj(x)1667 2529 y Fm(\))28 b Fl(\032)h Fi(B)1921 2493 y Fj(x)1916 2556 y(r)1955 2536 y Fd(0)1987 2529 y Fi(:)k Fm(This)h(implies)g(that)g(ev)m(ery)f(tile)g (in)h Fi(\036)p Fm(\()p Fi(F)3560 2493 y Fj(x)3610 2529 y Fm(\))g(is)150 2649 y(a)h(translate)g(b)m(y)h(an)f(elemen)m(t)h(of)g Fl(L)f Fm(of)h(a)f(tile)f(in)i Fi(F)2095 2612 y Fj(x)2145 2649 y Fi(;)f Fm(and)h(it)f(follo)m(ws)g(that)g(all)f(tiles)h(in)h Fi(x)f Fm(are)150 2768 y(translates)k(of)g(tiles)g(in)g Fi(F)1147 2732 y Fj(x)1197 2768 y Fi(;)f Fm(con)m(tradicting)h(that)g Fi(x)g Fm(should)g(ha)m(v)m(e)h(tiles)e(in)h(in\014nitely)g(man)m(y)150 2888 y(di\013eren)m(t)34 b(orien)m(tations.)p 1143 2898 60 80 v 349 3127 a(F)-8 b(or)32 b(a)h(tile)f Fi(T)46 b Fm(and)33 b Fi(n)28 b Fl(\025)g Fm(0)k(let)g Fl(P)1565 3142 y Fj(n)1620 3127 y Fm(\()p Fi(T)14 b Fm(\))32 b(b)s(e)h(the)g(set) f(of)h(admissable)g(patc)m(hes)h Fi(P)46 b Fm(for)33 b(whic)m(h)150 3247 y Fi(\036)209 3211 y Fj(n)264 3247 y Fm(\()p Fi(T)14 b Fm(\))33 b Fl(\032)i Fi(\036)618 3211 y Fj(n)672 3247 y Fm(\()p Fi(P)14 b Fm(\))37 b(and)g Fi(\036)1121 3211 y Fj(n)1176 3247 y Fm(\()p Fi(T)14 b Fm(\))33 b Fl(6\032)i Fi(\036)1530 3211 y Fj(n)1585 3247 y Fm(\()p Fi(P)1702 3211 y Fh(0)1729 3247 y Fm(\))i(for)g(an)m(y)g (prop)s(er)h(subpatc)m(h)h Fi(P)2982 3211 y Fh(0)3046 3247 y Fm(of)f Fi(P)s(:)e Fm(Then)i(eac)m(h)150 3366 y Fl(P)219 3381 y Fj(n)274 3366 y Fm(\()p Fi(T)14 b Fm(\))32 b(is)h(\014nite)h(and)g Fl(f)p Fi(T)14 b Fl(g)27 b Fm(=)h Fl(P)1378 3381 y Fg(0)1423 3366 y Fm(\()p Fi(T)14 b Fm(\))27 b Fl(\032)i(P)1775 3381 y Fg(1)1820 3366 y Fm(\()p Fi(T)14 b Fm(\))27 b Fl(\032)h(\001)17 b(\001)g(\001)e Fi(:)150 3546 y Fn(Lemma)50 b(9.)75 b Fm(F)-8 b(or)43 b(eac)m(h)h(tile)f Fi(T)57 b Fm(there)44 b(is)f(a)g(p)s(ositiv)m(e)g(in)m(teger)h Fi(N)2719 3561 y Fj(T)2825 3546 y Fm(suc)m(h)h(that)e Fl(P)3355 3561 y Fj(N)3418 3571 y Fe(T)3478 3546 y Fm(\()p Fi(T)14 b Fm(\))45 b(=)150 3665 y Fl(P)219 3680 y Fj(N)282 3690 y Fe(T)337 3680 y Fg(+1)443 3665 y Fm(\()p Fi(T)14 b Fm(\))28 b(=)g Fl(\001)17 b(\001)g(\001)e Fi(:)150 3845 y Fm(Pro)s(of.)64 b(Set)40 b Fl(P)8 b Fm(\()p Fi(T)14 b Fm(\))39 b(=)g Fl([)1108 3860 y Fj(n)p Fh(\025)p Fg(0)1265 3845 y Fl(P)1334 3860 y Fj(n)1388 3845 y Fm(\()p Fi(T)14 b Fm(\))40 b(and)g(let)g Fi(r)h(>)f Fm(0)p Fi(;)56 b Fn(y)40 b Fl(2)g Fk(E)2538 3809 y Fj(d)2624 3845 y Fm(suc)m(h)i(that)d Fi(B)3154 3860 y Fj(r)3198 3845 y Fm(\()p Fn(y)r Fm(\))f Fl(\032)i Fi(T)8 b(:)40 b Fm(By)150 3964 y(\014nite)45 b(lo)s(cal)e(complexit)m(y)-8 b(,)45 b Fl(P)8 b Fm(\()p Fi(T)14 b Fm(\))44 b(has)h(only)e(\014nitely)h(man)m(y)g(patc)m(hes)i (up)f(to)e(rigid)h(motion,)150 4084 y(since)34 b(ev)m(ery)f Fi(P)42 b Fl(2)28 b(P)8 b Fm(\()p Fi(T)14 b Fm(\))33 b(is)g(of)h(the)f(form)h(\()p Fi(T)1849 4048 y Fh(\016)1894 4084 y Fm(\))1933 4048 y Fj(x)2017 4084 y Fm(for)f(some)h(tiling)e Fi(x)27 b Fl(2)h Fi(X)r(:)349 4204 y Fm(If)41 b Fi(P)s(;)17 b(g)t(P)52 b Fl(2)41 b(P)912 4219 y Fj(n)967 4204 y Fm(\()p Fi(T)14 b Fm(\))40 b(for)h(some)g Fi(g)i Fl(2)e(G)1831 4219 y Fj(E)1939 4204 y Fm(with)g Fl(k)p Fi(g)t Fn(b)25 b Fl(\000)j Fn(b)p Fl(k)40 b Fi(<)g(K)7 b(r)43 b Fm(for)e(all)f Fn(b)g Fl(2)h Fi(P)54 b Fm(then)150 4323 y Fi(H)241 4287 y Fj(n)295 4323 y Fm(\()p Fi(g)t Fm(\))43 b(is)i(a)f(p)s(erio)s(d)g (for)h Fi(\036)1218 4287 y Fj(n)1273 4323 y Fm(\()p Fi(P)14 b Fm(\))44 b(whic)m(h)h(violates)e(Prop)s(osition)h(3.)77 b(Th)m(us)46 b(for)f(eac)m(h)g(patc)m(h)150 4443 y Fi(P)j Fl(2)34 b(P)8 b Fm(\()p Fi(T)14 b Fm(\))36 b(the)i(set)f(of)g Fi(g)g Fl(2)d(G)1321 4458 y Fj(E)1425 4443 y Fm(for)j(whic)m(h)h Fi(g)t(P)47 b Fl(2)34 b(P)8 b Fm(\()p Fi(T)14 b Fm(\))37 b(is)f(discrete)i(and)f(b)s(ounded,)i(hence)150 4562 y(\014nite.)44 b(It)33 b(follo)m(ws)g(that)g Fl(P)8 b Fm(\()p Fi(T)14 b Fm(\))33 b(is)h(\014nite,)f(whic)m(h)h(is)f(equiv)-6 b(alen)m(t)33 b(to)g(the)h(desired)g(result.)p 3559 4572 V 150 4742 a Fn(Lemma)46 b(10.)64 b Fm(Supp)s(ose)41 b Fl(P)1240 4757 y Fj(n)1295 4742 y Fm(\()p Fi(T)14 b Fm(\))38 b(=)h Fl(P)1668 4757 y Fj(n)p Fg(+1)1823 4742 y Fm(\()p Fi(T)14 b Fm(\))p Fi(:)39 b Fm(If)i Fi(x)d Fl(2)h Fi(X)47 b Fm(is)40 b(suc)m(h)h(that)f Fi(\036)3097 4706 y Fj(n)p Fg(+1)3252 4742 y Fm(\()p Fi(T)14 b Fm(\))38 b Fl(\032)h Fi(\036)p Fm(\()p Fi(x)p Fm(\))150 4861 y(then)34 b Fi(\036)436 4825 y Fj(n)491 4861 y Fm(\()p Fi(T)14 b Fm(\))27 b Fl(\032)h Fi(x:)150 5041 y Fm(Pro)s(of.)87 b(Let)48 b Fi(x)755 5005 y Fh(0)835 5041 y Fl(2)k Fi(X)j Fm(suc)m(h)50 b(that)d Fi(\036)1620 5005 y Fj(n)1675 5041 y Fm(\()p Fi(x)1771 5005 y Fh(0)1798 5041 y Fm(\))52 b(=)g Fi(x:)c Fm(Then)g Fi(\036)2484 5005 y Fj(n)p Fg(+1)2640 5041 y Fm(\()p Fi(T)14 b Fm(\))51 b Fl(\032)h Fi(\036)p Fm(\()p Fi(x)p Fm(\))g(=)g Fi(\036)3404 5005 y Fj(n)p Fg(+1)3560 5041 y Fm(\()p Fi(x)3656 5005 y Fh(0)3684 5041 y Fm(\))p Fi(;)150 5160 y Fm(hence)d(there)f(exists)f Fi(P)65 b Fl(2)51 b(P)1314 5175 y Fj(n)p Fg(+1)1469 5160 y Fm(\()p Fi(T)14 b Fm(\))47 b(suc)m(h)i(that)e Fi(P)65 b Fl(\032)51 b Fi(x)2449 5124 y Fh(0)2477 5160 y Fi(:)c Fm(Since)h Fi(P)65 b Fl(2)52 b(P)3142 5175 y Fj(n)3196 5160 y Fm(\()p Fi(T)14 b Fm(\))47 b(w)m(e)h(ha)m(v)m(e)150 5280 y Fi(\036)209 5244 y Fj(n)264 5280 y Fm(\()p Fi(T)14 b Fm(\))27 b Fl(\032)h Fi(\036)605 5244 y Fj(n)660 5280 y Fm(\()p Fi(P)14 b Fm(\))27 b Fl(\032)h Fi(\036)1007 5244 y Fj(n)1061 5280 y Fm(\()p Fi(x)1157 5244 y Fh(0)1185 5280 y Fm(\))f(=)h Fi(x:)p 1501 5290 V 1900 5519 a Fm(14)p eop %%Page: 15 16 15 15 bop 150 100 a Fn(Pro)s(of)39 b(of)e(Theorem)g(2.)44 b Fm(Set)33 b Fi(N)38 b Fm(=)28 b(max)p Fl(f)p Fi(N)1924 115 y Fj(T)2014 100 y Fm(:)g Fi(T)41 b Fl(2)28 b(Ag)p Fi(:)k Fm(Let)h Fi(x)27 b Fl(2)h Fi(X)r(;)k Fm(and)h(let)f Fi(x)3347 115 y Fg(1)3392 100 y Fi(;)17 b(x)3494 115 y Fg(2)3566 100 y Fl(2)28 b Fi(X)150 219 y Fm(b)s(e)34 b(an)m(y)f(tilings)f(suc)m(h)j(that)1448 339 y Fi(\036)p Fm(\()p Fi(x)1603 354 y Fg(1)1648 339 y Fm(\))27 b(=)h Fi(\036)1878 298 y Fj(N)7 b Fg(+1)2055 339 y Fm(\()p Fi(x)2151 354 y Fg(2)2196 339 y Fm(\))27 b(=)h Fi(x:)1121 b Fm(\(31\))150 528 y(W)-8 b(e)34 b(only)e(need)j(to)d(sho)m(w)i(that)f Fi(x)1414 543 y Fg(1)1486 528 y Fm(=)c Fi(\036)1651 492 y Fj(N)1727 528 y Fm(\()p Fi(x)1823 543 y Fg(2)1867 528 y Fm(\).)349 648 y(Let)34 b Fi(T)41 b Fl(2)29 b Fi(x)779 663 y Fg(2)856 648 y Fm(b)s(e)34 b(an)m(y)f(tile)g(and)g(let)g Fi(g)e Fl(2)d(G)1921 663 y Fj(E)2022 648 y Fm(suc)m(h)35 b(that)e Fi(g)t(T)46 b Fm(is)33 b(a)g(tile)g(in)g Fl(A)p Fm(.)43 b(Then)1085 887 y Fi(\036)1144 846 y Fj(N)7 b Fg(+1)1321 887 y Fm(\()p Fi(g)t(T)14 b Fm(\))26 b Fl(\032)j Fi(\036)1714 846 y Fj(N)7 b Fg(+1)1890 887 y Fm(\()p Fi(g)t(x)2038 902 y Fg(2)2082 887 y Fm(\))27 b(=)h Fi(\036)p Fm(\()p Fi(H)2442 846 y Fj(N)2518 887 y Fm(\()p Fi(g)t Fm(\))p Fi(x)2705 902 y Fg(1)2748 887 y Fm(\))p Fi(:)758 b Fm(\(32\))150 1126 y(By)32 b(Lemma)i(10,)e Fi(\036)881 1090 y Fj(N)957 1126 y Fm(\()p Fi(g)t(T)14 b Fm(\))27 b Fl(\032)h Fi(H)1382 1090 y Fj(N)1457 1126 y Fm(\()p Fi(g)t Fm(\))p Fi(x)1644 1141 y Fg(1)1688 1126 y Fi(;)k Fm(and)i(hence)h Fi(\036)2278 1090 y Fj(N)2354 1126 y Fm(\()p Fi(T)14 b Fm(\))27 b Fl(\032)h Fi(x)2693 1141 y Fg(1)2738 1126 y Fm(.)p 2825 1136 60 80 v 150 1545 a Fn(I)s(I)s(I.)40 b(References)166 1785 y Fm([G])49 b(C.)26 b(Go)s(o)s(dman-Strauss,)i(Matc)m(hing)e(rules)g(and)h (substitution)f(tilings,)g Fc(A)m(nnals)j(of)g(Math.)p Fm(,)349 1904 y Fn(147)34 b Fm(\(1998\))e(181-223.)41 2024 y([HRS])49 b(C.)33 b(Holton,)f(C.)h(Radin,)g(and)h(L.)f(Sadun,)h (in)g(preparation.)90 2144 y([LM])50 b(D.)40 b(Lind)h(and)f(B.)g (Marcus,)i Fc(A)m(n)h(Intr)-5 b(o)g(duction)42 b(to)g(Symb)-5 b(olic)42 b(Dynamics)g(and)g(Co)-5 b(ding)p Fm(,)349 2263 y(Cam)m(bridge)34 b(Univ)m(ersit)m(y)f(Press,)g(Cam)m(bridge,)h (1995.)153 2383 y([M])49 b(B.)35 b(Moss)m(\023)-47 b(e,)37 b(Puissances)h(de)e(mots)g(et)g(reconnaisabilit)m(\023)-47 b(e)36 b(des)g(p)s(oin)m(t)g(\014xes)g(d'une)h(substi-)349 2502 y(tution,)c Fc(The)-5 b(or.)35 b(Comp.)h(Sci.)43 b Fn(99)34 b Fm(no.)f(2)g(\(1992\))f(327-334.)38 2622 y([ORS])50 b(N.)33 b(Ormes,)i(C.)f(Radin,)g(and)g(L.)g(Sadun,)h(A)e (homeomorphism)i(in)m(v)-6 b(arian)m(t)34 b(for)g(substitu-)349 2741 y(tion)f(tiling)f(spaces,)i Fc(Ge)-5 b(ometriae)35 b(De)-5 b(dic)g(ata)33 b Fn(90)h Fm(\(2002\),)d(153-182.)176 2861 y([P])49 b(K.)34 b(P)m(etersen,)g(F)-8 b(actor)33 b(maps)h(b)s(et)m(w)m(een)h(tiling)e(dynamical)g(systems,)g Fc(F)-8 b(orum)38 b(Math.)44 b Fn(11)349 2980 y Fm(\(1999\))32 b(503-512.)171 3100 y([R])49 b(C.)d(Radin,)k(The)d(pin)m(wheel)h (tilings)d(of)i(the)f(plane,)k Fc(A)m(nnals)f(of)e(Math.)83 b Fn(139)48 b Fm(\(1994\),)349 3219 y(661-702.)66 3339 y([RS1])h(C.)32 b(Radin)h(and)g(L.)f(Sadun,)i(An)e(algebraic)g(in)m(v) -6 b(arian)m(t)32 b(for)h(substitution)g(tiling)e(systems,)349 3459 y Fc(Ge)-5 b(ometriae)35 b(De)-5 b(dic)g(ata)33 b Fn(73)g Fm(\(1998\),)f(21-37.)66 3578 y([RS2])49 b(C.)25 b(Radin)i(and)f(L.)f(Sadun,)j(Isomorphism)f(of)e(hierarc)m(hical)h (structures,)j Fc(Er)-5 b(go)g(dic)27 b(The)-5 b(ory)349 3698 y(Dynam.)47 b(Systems)33 b Fn(21)h Fm(\(2001\),)d(1239-1248.)79 3817 y([R)-11 b(W])50 b(C.)31 b(Radin)h(and)g(M.)f(W)-8 b(ol\013,)31 b(Space)h(tilings)f(and)g(lo)s(cal)g(isomorphism,)g Fc(Ge)-5 b(ometriae)33 b(De)-5 b(d-)349 3937 y(ic)g(ata)33 b Fn(42)g Fm(\(1992\),)f(355-360.)189 4056 y([S])49 b(B.)31 b(Solom)m(y)m(ak,)f(Nonp)s(erio)s(dicit)m(y)h(implies)g(unique)h(comp)s (osition)f(for)h(self-similar)g(trans-)349 4176 y(lationally)f (\014nite)j(tilings,)e Fc(Discr)-5 b(ete)35 b(Comput.)h(Ge)-5 b(om.)44 b Fn(20)34 b Fm(\(1998\),)d(265-279.)1900 5519 y(15)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0306061723140--