Content-Type: multipart/mixed; boundary="-------------9810300955749" This is a multi-part message in MIME format. ---------------9810300955749 Content-Type: text/plain; name="98-693.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="98-693.keywords" positive curvature, topological transitivity, topological mixing, geodesic flow, stochastic behavior ---------------9810300955749 Content-Type: application/postscript; name="bw.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="bw.ps" %!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: bw.dvi %%Pages: 20 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentPaperSizes: Letter %%EndComments %DVIPSCommandLine: dvips bw.dvi %DVIPSParameters: dpi=600, comments removed %DVIPSSource: TeX output 1998.09.29:1516 %%BeginProcSet: tex.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 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h(\014rst)g(p)s(erturb)g(the)g(round)g(metric)e(to)h(create)h(a)f(set)h (of)g Fo(")p Fu(-dense)0 219 y(h)m(yp)s(erb)s(olic)25 b(closed)g(orbits)f Fo(O)1114 234 y Fk(1)1159 219 y Fo(;)17 b(:)g(:)g(:)d(;)j(O)1457 234 y Fn(m)1556 219 y Fu(\(tangen)m(t)25 b(to)e(great)i(circles)f Fo(\015)2650 234 y Fk(1)2695 219 y Fo(;)17 b(:)g(:)g(:)d(;)j(\015)2969 234 y Fn(m)3043 219 y Fu(\))24 b(with)h(a)f(hetero)s(clinic)0 339 y(connection)37 b(b)s(et)m(w)m(een)g(eac)m(h)g(successiv)m(e)g(pair)f(of)g(orbits,)g (i.e.,)f(for)h(eac)m(h)h Fo(i)f Fu(there)g(exists)g(a)f(bi-in\014nite)0 458 y(orbit)j Fo(O)325 473 y Fn(i;i)p Fk(+1)549 458 y Fu(\(tangen)m(t)h(to)f(a)g(geo)s(desic)h Fo(c)1610 473 y Fn(i)1643 458 y Fu(\))f(that)g(is)h(bac)m(kw)m(ards)g(asymptotic)e (to)h Fo(O)3258 473 y Fn(i)3329 458 y Fu(and)h(forw)m(ards)0 578 y(asymptotic)29 b(to)h Fo(O)705 593 y Fn(i)p Fk(+1)839 578 y Fu(.)43 b(W)-8 b(e)30 b(then)h(e\013ect)g(small)e(lo)s(calized)h (metric)f(p)s(erturbations)i(to)f(break)h(all)e(these)0 697 y(hetero)s(clinic)39 b(connections)g(so)g(that)f(the)h(new)g (stable)g(and)g(unstable)g(manifolds)f(of)h(the)g(p)s(erturb)s(ed)0 817 y(h)m(yp)s(erb)s(olic)44 b(closed)h(geo)s(desics)g(\(whic)m(h)g (remain)e(h)m(yp)s(erb)s(olic\))h(in)m(tersect)h(transv)m(ersely)-8 b(.)76 b(It)44 b(follo)m(ws)0 936 y(that)39 b(this)g(p)s(erturb)s(ed)h (metric)e(has)h(an)g Fo(")p Fu(-dense)i(horsesho)s(e)f(\(lo)s(cally)d (maximal)f(h)m(yp)s(erb)s(olic)j(set\))g(in)0 1056 y(the)j(unit)f (tangen)m(t)h(bundle)h(and)f(th)m(us)g(will)f(ha)m(v)m(e)h(an)f(orbit)h (whic)m(h)g(is)f Fo(")p Fu(-dense)j(in)d(the)h(whole)g(unit)0 1176 y(tangen)m(t)36 b(bundle.)54 b(It)36 b(also)f(follo)m(ws)h(that)g (the)g(geo)s(desic)h(\015o)m(w)f(for)g(this)g(p)s(erturb)s(ed)i(metric) d(will)g(ha)m(v)m(e)0 1295 y(p)s(ositiv)m(e)e(top)s(ological)e(en)m (trop)m(y)-8 b(.)120 1479 y(By)29 b(carefully)i(iterating)e(the)i (construction)g(in)f(the)h(pro)s(of)f(of)h(Theorem)f(0.1,)g(w)m(e)h (exhibit)f(a)g(metric)0 1599 y(on)38 b(the)g(sphere)h Fo(S)701 1563 y Fk(3)783 1599 y Fu(that)f(is)f(close)h(to)f(the)h (round)h(metric)e(and)h(whose)g(geo)s(desic)g(\015o)m(w)g(has)h(an)e (orbit)0 1718 y(whose)d(closure)g(has)f(almost)f(full)h(measure.)44 b(More)34 b(precisely:)0 1902 y Fw(Theorem)47 b(0.2.)57 b Fj(Given)43 b Fo(")f(>)g Fu(0)p Fj(,)i(ther)-5 b(e)43 b(exists)f(a)h Fo(C)2070 1866 y Fi(1)2198 1902 y Fj(metric)g Fo(g)k Fj(on)c Fo(S)2832 1866 y Fk(3)2920 1902 y Fj(that)g(is)f(within) h Fo(")g Fj(of)g(the)0 2022 y(r)-5 b(ound)29 b(metric)g Fo(g)633 2037 y Fk(0)706 2022 y Fj(\(in)f(the)h Fo(C)1099 1986 y Fi(1)1212 2022 y Fj(top)-5 b(olo)g(gy\))29 b(with)g(the)f(pr)-5 b(op)g(erty)29 b(that)g(ther)-5 b(e)29 b(is)f(an)g(orbit)h(of)f(the)h (ge)-5 b(o)g(desic)0 2141 y(\015ow)35 b Fo(\036)270 2105 y Fn(t)270 2166 y(g)353 2141 y Fj(whose)g(closur)-5 b(e)36 b(has)f(\(normalize)-5 b(d\))35 b(Liouvil)5 b(le)36 b(me)-5 b(asur)g(e)36 b(at)f(le)-5 b(ast)36 b Fu(1)22 b Fm(\000)h Fo(")p Fj(.)120 2329 y Fu(It)44 b(is)i(not)f(imp)s(ossible)f(that)h (\(at)f(least)h(some)g(of)8 b(\))45 b(the)g(metrics)g(constructed)i(in) e(Theorem)g(0.2)0 2448 y(actually)e(ha)m(v)m(e)h(top)s(ologically)d (transitiv)m(e)i(geo)s(desic)i(\015o)m(ws.)76 b(Ho)m(w)m(ev)m(er,)46 b(w)m(e)e(do)g(not)g(ho)m(w)g(to)f(sho)m(w)0 2568 y(this.)120 2689 y(The)27 b(construction)g(used)h(to)e(pro)m(v)m(e)h(the)g(ab)s(o)m (v)m(e)g(theorems)f(requires)h(three)g(dimensions.)42 b(Ho)m(w)m(ev)m(er,)0 2808 y(on)g Fo(S)214 2772 y Fk(2)259 2808 y Fu(,)h(one)g(can)f(construct)h(a)e(metric)g(whose)i(geo)s (desics)f(appro)m(ximate)f(tra)6 b(jectories)42 b(of)g(the)g(w)m(ell)0 2928 y(kno)m(wn)e(stadium)f(billiard.)62 b(Recall)39 b(that)g(the)h(stadium)f(is)g(the)h Fo(C)2569 2892 y Fk(1)2654 2928 y Fu(con)m(v)m(ex)g(curv)m(e)g(formed)f(b)m(y)h(t)m(w)m (o)0 3047 y(semi)31 b(circles)h(joined)g(b)m(y)g(t)m(w)m(o)f(parallel)g (line)h(segmen)m(ts)f(\(see)h(Figure)g(1A\).)f(W)-8 b(e)31 b(can)h(appro)m(ximate)f(the)0 3167 y(stadium)38 b(b)m(y)i(a)f(table)g (whose)h(b)s(oundary)f(is)g(a)g(smo)s(oth)f(con)m(v)m(ex)i(curv)m(e,)h (and)f(then)g(form)e(a)h(smo)s(oth)0 3287 y(con)m(v)m(ex)33 b(surface)h(that)f(con)m(tains)g(parallel)f(copies)h(of)g(this)g(table) g(separated)g(b)m(y)g(a)g(narro)m(w)f(strip)h(with)0 3406 y(v)m(ery)g(strong)g(p)s(ositiv)m(e)g(curv)-6 b(ature,)34 b(as)f(sho)m(wn)h(in)f(Figure)h(1B.)120 3527 y(It)e(is)g(ob)m(vious)h (that)f(if)h(the)g(top)f(and)h(b)s(ottom)e(of)i(the)f(surface)i(are)f (made)f(close)h(enough)g(together,)0 3647 y(and)39 b(eac)m(h)g(of)f (them)f(appro)m(ximates)g(the)i(stadium)e(w)m(ell)h(enough,)i(long)d (geo)s(desic)i(segmen)m(ts)f(on)h(the)0 3766 y(surface)d(will)d (closely)h(follo)m(w)g(tra)6 b(jectories)34 b(from)g(the)g(stadium)g (billiard.)46 b(The)35 b(geo)s(desic)g(will)e(switc)m(h)0 3886 y(b)s(et)m(w)m(een)38 b(the)g(top)e(and)i(the)f(b)s(ottom)e(of)i (the)g(surface)i(eac)m(h)e(time)f(the)h(billiard)f(tra)6 b(jectory)37 b(b)s(ounces)0 4005 y(o\013)c(the)g(edge)g(of)g(the)g (table.)44 b(The)33 b(stadium)f(billiard)g(is)h(ergo)s(dic,)f(and)h (hence)i(there)e(is)g(a)f(dense)i(orbit)0 4125 y(of)40 b(the)g(billiard)f(\015o)m(w.)64 b(W)-8 b(e)40 b(can)g(construct)h(our) f(surface)h(so)f(that)f(its)h(geo)s(desic)g(\015o)m(w)g(has)g(an)g (orbit)0 4244 y(that)c(appro)m(ximates)e(as)i(long)g(a)g(piece)g(of)g (this)g(dense)i(orbit)d(as)h(closely)g(as)g(w)m(e)g(wish.)52 b(Th)m(us)37 b(w)m(e)g(can)0 4364 y(pro)s(duce)i(a)f(metric)f(on)h Fo(S)997 4328 y Fk(2)1080 4364 y Fu(with)g(an)g Fo(")p Fu(-dense)i(orbit)e(of)g(the)h(geo)s(desic)f(\015o)m(w)h(for)f(an)m(y)g Fo(")e(>)h Fu(0.)58 b(These)0 4484 y(metrics)46 b(clearly)f(ha)m(v)m(e) i(non-negativ)m(e)g(curv)-6 b(ature.)84 b(Ho)m(w)m(ev)m(er,)49 b(an)e(easy)f(p)s(erturbation)g(argumen)m(t)0 4603 y(giv)m(es)d (strictly-con)m(v)m(ex)f(surfaces)i(\(metrics)e(with)g(p)s(ositiv)m(e)g (curv)-6 b(ature\))43 b(with)g(an)g Fo(")p Fu(-dense)i(orbit)d(of)0 4723 y(the)34 b(geo)s(desic)f(\015o)m(w)h(for)f(an)m(y)h Fo(")28 b(>)g Fu(0.)1140 5400 y @beginspecial 67 @llx 609 @lly 295 @urx 688 @ury 1620 @rwi 540 @rhi @setspecial %%BeginDocument: stadium_bill.ps % % Due to bugs in Transcript, the 'PS-Adobe-' stuff is omitted from line 1 % /tgifdict 132 dict def tgifdict begin % % Using a zero value radius for an ellipse or an arc would result % in a non-invertible CTM matrix which causes problem when this % when this PostScript is wrapped inside other routines, such as % the multi.ps package from % ftp.ucc.su.oz.au:/pub/ps_printing/multi. You can overcome such % error by uncommenting the sole line of the procedure below: % /tgif_min_radius { % dup 0.01 lt { pop 0.01 } if } bind def /tgifellipsedict 6 dict def tgifellipsedict /mtrx matrix put /tgifellipse { tgifellipsedict begin /yrad exch def /xrad exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y translate xrad yrad scale 0 0 1 0 360 arc savematrix setmatrix end } def /tgifarrowtipdict 8 dict def tgifarrowtipdict /mtrx matrix put /tgifarrowtip { tgifarrowtipdict begin /dy exch def /dx exch def /h exch def /w exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y translate dy dx atan rotate 0 0 moveto w neg h lineto w neg h neg lineto savematrix setmatrix end } def /tgifarcdict 8 dict def tgifarcdict /mtrx matrix put /tgifarcn { tgifarcdict begin /endangle exch def /startangle exch def /yrad exch def /xrad exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y translate xrad yrad scale 0 0 1 startangle endangle arc savematrix setmatrix end } def /tgifarc { tgifarcdict begin /endangle exch def /startangle exch def /yrad exch def /xrad exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y translate xrad yrad scale 0 0 1 startangle endangle arcn savematrix setmatrix end } def /tgifsetuserscreendict 22 dict def tgifsetuserscreendict begin /tempctm matrix def /temprot matrix def /tempscale matrix def /concatprocs { /proc2 exch cvlit def /proc1 exch cvlit def /newproc proc1 length proc2 length add array def newproc 0 proc1 putinterval newproc proc1 length proc2 putinterval newproc cvx } def /resmatrix matrix def /findresolution { 72 0 resmatrix defaultmatrix dtransform /yres exch def /xres exch def xres dup mul yres dup mul add sqrt } def end /tgifsetuserscreen { tgifsetuserscreendict begin /spotfunction exch def /screenangle exch def /cellsize exch def /m tempctm currentmatrix def /rm screenangle temprot rotate def /sm cellsize dup tempscale scale def sm rm m m concatmatrix m concatmatrix pop 1 0 m dtransform /y1 exch def /x1 exch def /veclength x1 dup mul y1 dup mul add sqrt def /frequency findresolution veclength div def /newscreenangle y1 x1 atan def m 2 get m 1 get mul m 0 get m 3 get mul sub 0 gt {{neg} /spotfunction load concatprocs /spotfunction exch def } if frequency newscreenangle /spotfunction load setscreen end } def /tgifsetpatterndict 18 dict def tgifsetpatterndict begin /bitison { /ybit exch def /xbit exch def /bytevalue bstring ybit bwidth mul xbit 8 idiv add get def /mask 1 7 xbit 8 mod sub bitshift def bytevalue mask and 0 ne } def end /tgifbitpatternspotfunction { tgifsetpatterndict begin /y exch def /x exch def /xindex x 1 add 2 div bpside mul cvi def /yindex y 1 add 2 div bpside mul cvi def xindex yindex bitison { /onbits onbits 1 add def 1 } { /offbits offbits 1 add def 0 } ifelse end } def /tgifsetpattern { tgifsetpatterndict begin /cellsz exch def /angle exch def /bwidth exch def /bpside exch def /bstring exch def /onbits 0 def /offbits 0 def cellsz angle /tgifbitpatternspotfunction load tgifsetuserscreen {} settransfer offbits offbits onbits add div setgray end } def /tgifxpmdict 4 dict def /tgifbwpicstr 1 string def /tgifcolorpicstr 3 string def /tgifsetpixels { tgifxpmdict begin /pixels exch def end } def /tgifsetpix { tgifxpmdict begin pixels 3 1 roll putinterval end } def /tgifbwspot { tgifxpmdict begin /index exch def tgifbwpicstr 0 pixels index 3 mul 3 getinterval aload pop 255 mul .114 mul exch 255 mul .587 mul add exch 255 mul .299 mul add cvi put tgifbwpicstr end } def /tgifcolorspot { tgifxpmdict begin /index exch def pixels index 3 mul 3 getinterval aload pop 255 mul cvi tgifcolorpicstr 2 3 -1 roll put 255 mul cvi tgifcolorpicstr 1 3 -1 roll put 255 mul cvi tgifcolorpicstr 0 3 -1 roll put tgifcolorpicstr end } def /tgifnewcolorspot { tgifxpmdict begin /index exch def pixels index 3 mul 3 getinterval aload pop setrgbcolor end } def /tgifcolordict 4 dict def /colorimage where { pop } { /colorimage { tgifcolordict begin pop pop pop pop pop /ih exch def /iw exch def /x 0 def /y 0 def 1 1 ih { pop 1 1 iw { pop currentfile tgifbwpicstr readhexstring pop 0 get tgifnewcolorspot x y moveto 1 0 rlineto 0 1 rlineto -1 0 rlineto closepath fill /x x 1 add def } for /y y 1 add def /x 0 def } for end } def } ifelse /tgifpatdict 10 dict def /tgifpatbyte { currentdict /retstr get exch pat i cellsz mod get put } def /tgifpatproc { 0 1 widthlim {tgifpatbyte} for retstr /i i 1 add def } def /tgifpatfill { tgifpatdict begin /h exch def /w exch def /lty exch def /ltx exch def /cellsz exch def /pat exch def /widthlim w cellsz div cvi 1 sub def /retstr widthlim 1 add string def /i 0 def ltx lty translate w h true [1 0 0 1 0 0] {tgifpatproc} imagemask ltx neg lty neg translate end } def /pat1 def /pat2 <0000000000000000> def /pat3 <8000000008000000> def /pat4 <8800000022000000> def /pat5 <8800220088002200> def /pat6 <8822882288228822> def /pat7 def /pat8 <77dd77dd77dd77dd> def /pat9 <77ffddff77ffddff> def /pat10 <77ffffff77ffffff> def /pat11 <7fffffff7fffffff> def /pat12 <8040200002040800> def /pat13 <40a00000040a0000> def /pat14 def /pat15 def /pat16 def /pat17 <038448300c020101> def /pat18 <081c22c180010204> def /pat19 <8080413e080814e3> def /pat20 <8040201008040201> def /pat21 <8844221188442211> def /pat22 <77bbddee77bbddee> def /pat23 def /pat24 <7fbfdfeff7fbfdfe> def /pat25 <3e1f8fc7e3f1f87c> def /pat26 <0102040810204080> def /pat27 <1122448811224488> def /pat28 def /pat29 <83070e1c3870e0c1> def /pat30 def /pat31 <7cf8f1e3c78f1f3e> def /tgifcentertext { dup stringwidth pop 2 div neg 0 rmoveto } def /tgifrighttext { dup stringwidth pop neg 0 rmoveto } def /tgifreencsmalldict 12 dict def /tgifReEncodeSmall { tgifreencsmalldict begin /newcodesandnames exch def /newfontname exch def /basefontname exch def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict { exch dup /FID ne { dup /Encoding eq { exch dup length array copy newfont 3 1 roll put } { exch newfont 3 1 roll put } ifelse } { pop pop } ifelse } forall newfont /FontName newfontname put newcodesandnames aload pop newcodesandnames length 2 idiv { newfont /Encoding get 3 1 roll put} repeat newfontname newfont definefont pop end } def /tgifgray { 8 1 0 72 300 32 div div tgifsetpattern } bind def /tgifboxdict 6 dict def /tgifboxstroke { tgifboxdict begin /pat def /w def /y2 exch def /x2 exch def /y1 exch def /x1 exch def 1.415 setmiterlimit w 1 eq { w setlinewidth } if pat pat1 ne pat pat2 ne and { gsave pat tgifgray } if newpath x1 y1 moveto x2 y1 lineto x2 y2 lineto x1 y2 lineto closepath pat pat2 eq { 1 setgray stroke 0 setgray } { stroke } ifelse pat pat1 ne pat pat2 ne and { grestore } if w 1 eq { 1 setlinewidth } if 1 setmiterlimit end } def /tgifboxfill { tgifboxdict begin /pat def /y2 exch def /x2 exch def /y1 exch def /x1 exch def pat pat1 ne pat pat2 ne and { gsave pat tgifgray } if newpath x1 y1 moveto x2 y1 lineto x2 y2 lineto x1 y2 lineto closepath pat pat2 eq { 1 setgray fill 0 setgray } { fill } ifelse pat pat1 ne pat pat2 ne and { grestore } if end } def end tgifdict begin /tgifsavedpage save def 1 setmiterlimit 1 setlinewidth 72 0 mul 72 11.00 mul translate 72 128 div 100 mul 100 div dup neg scale gsave % ARC gsave 5 setlinewidth newpath 456 254 66 67 90 -90 tgifarc stroke 1 setlinewidth grestore % POLY/OPEN-SPLINE gsave 5 setlinewidth newpath 458 187 moveto 187 187 lineto stroke 1 setlinewidth grestore % POLY/OPEN-SPLINE gsave 5 setlinewidth newpath 457 321 moveto 187 321 lineto stroke 1 setlinewidth grestore % ARC gsave 5 setlinewidth newpath 189 254 66 67 -90 -270 tgifarc stroke 1 setlinewidth grestore % POLY/OPEN-SPLINE gsave 2 setlinewidth [12 4 4 4 4 4] 0 setdash newpath 128 285 moveto -40 84 atan dup cos 10 mul 212 exch sub exch sin 10 mul 245 exch sub lineto stroke [] 0 setdash 1 setlinewidth grestore gsave newpath 212 245 10 4 84 -40 tgifarrowtip closepath fill grestore % POLY/OPEN-SPLINE gsave 2 setlinewidth [12 4 4 4 4 4] 0 setdash newpath 203 250 moveto 321 188 lineto 28 50 atan dup cos 10 mul 371 exch sub exch sin 10 mul 216 exch sub lineto stroke [] 0 setdash 1 setlinewidth grestore gsave newpath 371 216 10 4 50 28 tgifarrowtip closepath fill grestore % POLY/OPEN-SPLINE gsave 2 setlinewidth [12 4 4 4 4 4] 0 setdash newpath 364 212 moveto 507 297 lineto -77 -47 atan dup cos 10 mul 460 exch sub exch sin 10 mul 220 exch sub lineto stroke [] 0 setdash 1 setlinewidth grestore gsave newpath 460 220 10 4 -47 -77 tgifarrowtip closepath fill grestore % POLY/OPEN-SPLINE gsave 2 setlinewidth [12 4 4 4 4 4] 0 setdash newpath 463 225 moveto 441 186 lineto 110 -58 atan dup cos 10 mul 383 exch sub exch sin 10 mul 296 exch sub lineto stroke [] 0 setdash 1 setlinewidth grestore gsave newpath 383 296 10 4 -58 110 tgifarrowtip closepath fill grestore % POLY/OPEN-SPLINE gsave 2 setlinewidth [12 4 4 4 4 4] 0 setdash newpath 386 290 moveto 369 321 lineto -93 -71 atan dup cos 10 mul 298 exch sub exch sin 10 mul 228 exch sub lineto stroke [] 0 setdash 1 setlinewidth grestore gsave newpath 298 228 10 4 -71 -93 tgifarrowtip closepath fill grestore % POLY/OPEN-SPLINE gsave 2 setlinewidth [12 4 4 4 4 4] 0 setdash newpath 301 232 moveto 269 187 lineto stroke [] 0 setdash 1 setlinewidth grestore % POLY/OPEN-SPLINE gsave 2 setlinewidth [12 4 4 4 4 4] 0 setdash newpath 270 188 moveto 98 -59 atan dup cos 10 mul 211 exch sub exch sin 10 mul 286 exch sub lineto stroke [] 0 setdash 1 setlinewidth grestore gsave newpath 211 286 10 4 -59 98 tgifarrowtip closepath fill grestore % POLY/OPEN-SPLINE gsave 2 setlinewidth [12 4 4 4 4 4] 0 setdash newpath 216 278 moveto 192 319 lineto 151 255 lineto -8 -6 atan dup cos 10 mul 145 exch sub exch sin 10 mul 247 exch sub lineto stroke [] 0 setdash 1 setlinewidth grestore gsave newpath 145 247 10 4 -6 -8 tgifarrowtip closepath fill grestore % POLY/OPEN-SPLINE gsave 2 setlinewidth [12 4 4 4 4 4] 0 setdash newpath 150 254 moveto 130 221 lineto stroke [] 0 setdash 1 setlinewidth grestore grestore tgifsavedpage restore end showpage %MatchingCreationDate: Tue Apr 2 13:00:43 1996 %%EndDocument @endspecial 1929 5639 a Fs(3)p eop %%Page: 4 4 4 3 bop 1226 188 a Fv(Figure)38 b(1A.)f(St)-7 b(adium)39 b(Billiard)1140 812 y @beginspecial 78 @llx 653 @lly 242 @urx 706 @ury 1620 @rwi 540 @rhi @setspecial %%BeginDocument: double_stadium.eps % % Due to bugs in Transcript, the 'PS-Adobe-' stuff is omitted from line 1 % /tgifdict 132 dict def tgifdict begin % % Using a zero value radius for an ellipse or an arc would result % in a non-invertible CTM matrix which causes problem when this % when this PostScript is wrapped inside other routines, such as % the multi.ps package from % ftp.ucc.su.oz.au:/pub/ps_printing/multi. You can overcome such % error by uncommenting the sole line of the procedure below: % /tgif_min_radius { % dup 0.01 lt { pop 0.01 } if } bind def /tgifellipsedict 6 dict def tgifellipsedict /mtrx matrix put /tgifellipse { tgifellipsedict begin /yrad exch def /xrad exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y translate xrad yrad scale 0 0 1 0 360 arc savematrix setmatrix end } def /tgifarrowtipdict 8 dict def tgifarrowtipdict /mtrx matrix put /tgifarrowtip { tgifarrowtipdict begin /dy exch def /dx exch def /h exch def /w exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y translate dy dx atan rotate 0 0 moveto w neg h lineto w neg h neg lineto savematrix setmatrix end } def /tgifarcdict 8 dict def tgifarcdict /mtrx matrix put /tgifarcn { tgifarcdict begin /endangle exch def /startangle exch def /yrad exch def /xrad exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y translate xrad yrad scale 0 0 1 startangle endangle arc savematrix setmatrix end } def /tgifarc { tgifarcdict begin /endangle exch def /startangle exch def /yrad exch def /xrad exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y translate xrad yrad scale 0 0 1 startangle endangle arcn savematrix setmatrix end } def /tgifsetuserscreendict 22 dict def tgifsetuserscreendict begin /tempctm matrix def /temprot matrix def /tempscale matrix def /concatprocs { /proc2 exch cvlit def /proc1 exch cvlit def /newproc proc1 length proc2 length add array def newproc 0 proc1 putinterval newproc proc1 length proc2 putinterval newproc cvx } def /resmatrix matrix def /findresolution { 72 0 resmatrix defaultmatrix dtransform /yres exch def /xres exch def xres dup mul yres dup mul add sqrt } def end /tgifsetuserscreen { tgifsetuserscreendict begin /spotfunction exch def /screenangle exch def /cellsize exch def /m tempctm currentmatrix def /rm screenangle temprot rotate def /sm cellsize dup tempscale scale def sm rm m m concatmatrix m concatmatrix pop 1 0 m dtransform /y1 exch def /x1 exch def /veclength x1 dup mul y1 dup mul add sqrt def /frequency findresolution veclength div def /newscreenangle y1 x1 atan def m 2 get m 1 get mul m 0 get m 3 get mul sub 0 gt {{neg} /spotfunction load concatprocs /spotfunction exch def } if frequency newscreenangle /spotfunction load setscreen end } def /tgifsetpatterndict 18 dict def tgifsetpatterndict begin /bitison { /ybit exch def /xbit exch def /bytevalue bstring ybit bwidth mul xbit 8 idiv add get def /mask 1 7 xbit 8 mod sub bitshift def bytevalue mask and 0 ne } def end /tgifbitpatternspotfunction { tgifsetpatterndict begin /y exch def /x exch def /xindex x 1 add 2 div bpside mul cvi def /yindex y 1 add 2 div bpside mul cvi def xindex yindex bitison { /onbits onbits 1 add def 1 } { /offbits offbits 1 add def 0 } ifelse end } def /tgifsetpattern { tgifsetpatterndict begin /cellsz exch def /angle exch def /bwidth exch def /bpside exch def /bstring exch def /onbits 0 def /offbits 0 def cellsz angle /tgifbitpatternspotfunction load tgifsetuserscreen {} settransfer offbits offbits onbits add div setgray end } def /tgifxpmdict 4 dict def /tgifbwpicstr 1 string def /tgifcolorpicstr 3 string def /tgifsetpixels { tgifxpmdict begin /pixels exch def end } def /tgifsetpix { tgifxpmdict begin pixels 3 1 roll putinterval end } def /tgifbwspot { tgifxpmdict begin /index exch def tgifbwpicstr 0 pixels index 3 mul 3 getinterval aload pop 255 mul .114 mul exch 255 mul .587 mul add exch 255 mul .299 mul add cvi put tgifbwpicstr end } def /tgifcolorspot { tgifxpmdict begin /index exch def pixels index 3 mul 3 getinterval aload pop 255 mul cvi tgifcolorpicstr 2 3 -1 roll put 255 mul cvi tgifcolorpicstr 1 3 -1 roll put 255 mul cvi tgifcolorpicstr 0 3 -1 roll put tgifcolorpicstr end } def /tgifnewcolorspot { tgifxpmdict begin /index exch def pixels index 3 mul 3 getinterval aload pop setrgbcolor end } def /tgifcolordict 4 dict def /colorimage where { pop } { /colorimage { tgifcolordict begin pop pop pop pop pop /ih exch def /iw exch def /x 0 def /y 0 def 1 1 ih { pop 1 1 iw { pop currentfile tgifbwpicstr readhexstring pop 0 get tgifnewcolorspot x y moveto 1 0 rlineto 0 1 rlineto -1 0 rlineto closepath fill /x x 1 add def } for /y y 1 add def /x 0 def } for end } def } ifelse /tgifpatdict 10 dict def /tgifpatbyte { currentdict /retstr get exch pat i cellsz mod get put } def /tgifpatproc { 0 1 widthlim {tgifpatbyte} for retstr /i i 1 add def } def /tgifpatfill { tgifpatdict begin /h exch def /w exch def /lty exch def /ltx exch def /cellsz exch def /pat exch def /widthlim w cellsz div cvi 1 sub def /retstr widthlim 1 add string def /i 0 def ltx lty translate w h true [1 0 0 1 0 0] {tgifpatproc} imagemask ltx neg lty neg translate end } def /pat1 def /pat2 <0000000000000000> def /pat3 <8000000008000000> def /pat4 <8800000022000000> def /pat5 <8800220088002200> def /pat6 <8822882288228822> def /pat7 def /pat8 <77dd77dd77dd77dd> def /pat9 <77ffddff77ffddff> def /pat10 <77ffffff77ffffff> def /pat11 <7fffffff7fffffff> def /pat12 <8040200002040800> def /pat13 <40a00000040a0000> def /pat14 def /pat15 def /pat16 def /pat17 <038448300c020101> def /pat18 <081c22c180010204> def /pat19 <8080413e080814e3> def /pat20 <8040201008040201> def /pat21 <8844221188442211> def /pat22 <77bbddee77bbddee> def /pat23 def /pat24 <7fbfdfeff7fbfdfe> def /pat25 <3e1f8fc7e3f1f87c> def /pat26 <0102040810204080> def /pat27 <1122448811224488> def /pat28 def /pat29 <83070e1c3870e0c1> def /pat30 def /pat31 <7cf8f1e3c78f1f3e> def /tgifcentertext { dup stringwidth pop 2 div neg 0 rmoveto } def /tgifrighttext { dup stringwidth pop neg 0 rmoveto } def /tgifreencsmalldict 12 dict def /tgifReEncodeSmall { tgifreencsmalldict begin /newcodesandnames exch def /newfontname exch def /basefontname exch def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict { exch dup /FID ne { dup /Encoding eq { exch dup length array copy newfont 3 1 roll put } { exch newfont 3 1 roll put } ifelse } { pop pop } ifelse } forall newfont /FontName newfontname put newcodesandnames aload pop newcodesandnames length 2 idiv { newfont /Encoding get 3 1 roll put} repeat newfontname newfont definefont pop end } def /tgifgray { 8 1 0 72 300 32 div div tgifsetpattern } bind def /tgifboxdict 6 dict def /tgifboxstroke { tgifboxdict begin /pat def /w def /y2 exch def /x2 exch def /y1 exch def /x1 exch def 1.415 setmiterlimit w 1 eq { w setlinewidth } if pat pat1 ne pat pat2 ne and { gsave pat tgifgray } if newpath x1 y1 moveto x2 y1 lineto x2 y2 lineto x1 y2 lineto closepath pat pat2 eq { 1 setgray stroke 0 setgray } { stroke } ifelse pat pat1 ne pat pat2 ne and { grestore } if w 1 eq { 1 setlinewidth } if 1 setmiterlimit end } def /tgifboxfill { tgifboxdict begin /pat def /y2 exch def /x2 exch def /y1 exch def /x1 exch def pat pat1 ne pat pat2 ne and { gsave pat tgifgray } if newpath x1 y1 moveto x2 y1 lineto x2 y2 lineto x1 y2 lineto closepath pat pat2 eq { 1 setgray fill 0 setgray } { fill } ifelse pat pat1 ne pat pat2 ne and { grestore } if end } def end tgifdict begin /tgifsavedpage save def 1 setmiterlimit 1 setlinewidth 72 0 mul 72 11.00 mul translate 72 128 div 100 mul 100 div dup neg scale gsave % ARC gsave 2 setlinewidth newpath 371 191 54 28 -87 96 tgifarcn stroke 1 setlinewidth grestore % ARC gsave 2 setlinewidth newpath 386 196 43 38 -87 96 tgifarcn stroke 1 setlinewidth grestore % BOX gsave pat5 8 1 0 72 300 32 div div tgifsetpattern newpath 202 176 moveto 390 176 lineto 390 241 lineto 202 241 lineto closepath fill grestore % POLY/OPEN-SPLINE gsave 4 setlinewidth newpath 202 175 moveto 392 175 lineto stroke 1 setlinewidth grestore % ARC gsave pat5 8 1 0 72 300 32 div div tgifsetpattern newpath 388 208 moveto 388 208 39 33 90 -88 tgifarc 388 208 lineto fill grestore gsave 4 setlinewidth newpath 388 208 39 33 90 -88 tgifarc stroke 1 setlinewidth grestore % POLY/OPEN-SPLINE gsave 4 setlinewidth newpath 390 242 moveto 200 242 lineto stroke 1 setlinewidth grestore % ARC gsave pat5 8 1 0 72 300 32 div div tgifsetpattern newpath 204 208 moveto 204 208 39 33 -90 -268 tgifarc 204 208 lineto fill grestore gsave 4 setlinewidth newpath 204 208 39 33 -90 -268 tgifarc stroke 1 setlinewidth grestore % ARC gsave 2 setlinewidth newpath 172 209 24 26 -93 -276 tgifarc stroke 1 setlinewidth grestore % ARC gsave 2 setlinewidth newpath 190 217 33 22 -93 -276 tgifarc stroke 1 setlinewidth grestore % ARC gsave 2 setlinewidth newpath 406 214 17 23 -87 96 tgifarcn stroke 1 setlinewidth grestore % ARC gsave 2 setlinewidth newpath 259 232 15 11 -93 -276 tgifarc stroke 1 setlinewidth grestore % ARC gsave 2 setlinewidth newpath 213 232 15 11 -93 -276 tgifarc stroke 1 setlinewidth grestore % ARC gsave 2 setlinewidth newpath 233 232 15 9 -93 -276 tgifarc stroke 1 setlinewidth grestore % ARC gsave 2 setlinewidth newpath 282 232 15 9 -93 -276 tgifarc stroke 1 setlinewidth grestore % ARC gsave 2 setlinewidth newpath 309 232 15 9 -93 -276 tgifarc stroke 1 setlinewidth grestore % ARC gsave 2 setlinewidth newpath 336 232 15 9 -93 -276 tgifarc stroke 1 setlinewidth grestore % ARC gsave 2 setlinewidth newpath 365 232 15 9 -93 -276 tgifarc stroke 1 setlinewidth grestore % ARC gsave 2 setlinewidth newpath 389 230 15 11 -93 -276 tgifarc stroke 1 setlinewidth grestore % ARC gsave 2 setlinewidth newpath 405 222 12 17 -93 -276 tgifarc stroke 1 setlinewidth grestore % ARC gsave 2 setlinewidth newpath 204 225 22 17 -93 -276 tgifarc stroke 1 setlinewidth grestore % BOX gsave pat6 8 1 0 72 300 32 div div tgifsetpattern newpath 182 157 moveto 370 157 lineto 370 222 lineto 182 222 lineto closepath fill grestore % POLY/OPEN-SPLINE gsave 4 setlinewidth newpath 182 156 moveto 372 156 lineto stroke 1 setlinewidth grestore % ARC gsave pat6 8 1 0 72 300 32 div div tgifsetpattern newpath 368 189 moveto 368 189 39 33 90 -88 tgifarc 368 189 lineto fill grestore gsave 4 setlinewidth newpath 368 189 39 33 90 -88 tgifarc stroke 1 setlinewidth grestore % POLY/OPEN-SPLINE gsave 4 setlinewidth newpath 370 223 moveto 180 223 lineto stroke 1 setlinewidth grestore % ARC gsave pat6 8 1 0 72 300 32 div div tgifsetpattern newpath 184 189 moveto 184 189 39 33 -90 -268 tgifarc 184 189 lineto fill grestore gsave 4 setlinewidth newpath 184 189 39 33 -90 -268 tgifarc stroke 1 setlinewidth grestore % ARC gsave 2 setlinewidth newpath 155 199 12 29 -93 -276 tgifarc stroke 1 setlinewidth grestore % ARC gsave 2 setlinewidth newpath 401 222 20 16 -87 96 tgifarcn stroke 1 setlinewidth grestore grestore tgifsavedpage restore end %MatchingCreationDate: Thu Dec 11 13:56:11 1997 %%EndDocument @endspecial 1338 1201 a(Figure)f(1B.)f(Our)h(Surf)-9 b(a)n(ce)120 1489 y Fu(W)h(e)36 b(thank)g(Charles)h(Pugh)g(for)f(his)h (in)m(terest)g(in)f(this)h(w)m(ork)f(and)h(for)f(con)m(vincing)h(us)g (to)f(impro)m(v)m(e)0 1608 y(our)d(exp)s(osition)g(in)g(Section)h(4.) 903 1812 y Fw(1.)55 b(Preliminaries)36 b(from)h(Dynamical)f(Systems)120 1991 y Fu(A)27 b(con)m(tin)m(uous)i(\015o)m(w)f Fo(\036)976 1955 y Fn(t)1039 1991 y Fu(on)f(a)h(compact)e(metric)h(space)i Fo(X)34 b Fu(is)28 b Fj(top)-5 b(olo)g(gic)g(al)5 b(ly)31 b(tr)-5 b(ansitive)26 b Fu(if)i(giv)m(en)f(an)m(y)0 2111 y(t)m(w)m(o)37 b(non)g(empt)m(y)f(op)s(en)h(sets)g Fo(U;)17 b(V)55 b Fm(\032)34 b Fo(X)44 b Fu(,)37 b(there)g(exists)g Fo(T)47 b(>)34 b Fu(0)j(suc)m(h)h(that)e Fo(\036)3014 2075 y Fn(T)3077 2111 y Fu(\()p Fo(U)11 b Fu(\))24 b Fm(\\)h Fo(V)56 b Fm(6)p Fu(=)34 b Fm(;)p Fu(.)54 b(This)0 2230 y(is)36 b(equiv)-6 b(alen)m(t)37 b(to)f(the)g(existence)i(of)e (dense)i(orbit)e(of)h Fo(\036)2121 2194 y Fn(t)2156 2230 y Fu(.)54 b(W)-8 b(e)37 b(sa)m(y)f(that)g(the)h(\015o)m(w)f(is)h Fo(")p Fu(-)p Fj(top)-5 b(olo)g(gic)g(al)5 b(ly)0 2350 y(tr)-5 b(ansitive)39 b Fu(if)h(giv)m(en)g(an)m(y)h(t)m(w)m(o)f(op)s (en)h(sets)f Fo(U;)17 b(V)61 b Fm(\032)40 b Fo(X)48 b Fu(whic)m(h)41 b(con)m(tain)f(balls)g(of)h(diameter)e Fo(")p Fu(,)j(there)0 2469 y(exists)37 b Fo(T)48 b(>)34 b Fu(0)j(suc)m(h)h(that)e Fo(\036)1089 2433 y Fn(T)1153 2469 y Fu(\()p Fo(U)11 b Fu(\))23 b Fm(\\)i Fo(V)56 b Fm(6)p Fu(=)35 b Fm(;)p Fu(.)54 b(This)38 b(is)e(equiv)-6 b(alen)m(t)37 b(to)g(the)g(existence)h(of)f(an)g Fo(")p Fu(-dense)0 2589 y(orbit)c(of)g Fo(\036)416 2553 y Fn(t)452 2589 y Fu(,)g(i.e.,)f(there)i(exists)f(an)g(orbit)g(whic)m(h)h(in)m (tersects)g(ev)m(ery)g(ball)e(of)i(diameter)e(at)h(least)g Fo(")p Fu(.)120 2708 y(A)f(top)s(ological)f(prop)s(ert)m(y)i(of)g(a)f (\015o)m(w)h(whic)m(h)h(is)f(stronger)g(than)g(top)s(ological)e (transitivit)m(y)g(is)h(top)s(o-)0 2828 y(logical)h(mixing.)43 b(A)34 b(con)m(tin)m(uous)h(\015o)m(w)f Fo(\036)1555 2792 y Fn(t)1625 2828 y Fu(on)f(a)h(compact)f(metric)g(space)i Fo(X)41 b Fu(is)34 b Fj(top)-5 b(olo)g(gic)g(al)5 b(ly)36 b(mixing)0 2948 y Fu(if)30 b(giv)m(en)f(an)m(y)h(t)m(w)m(o)f(non)h (empt)m(y)f(op)s(en)h(sets)g Fo(U;)17 b(V)49 b Fm(\032)29 b Fo(X)36 b Fu(,)30 b(there)h(exists)e Fo(T)42 b(>)28 b Fu(0)h(suc)m(h)i(that)f(for)f(all)g Fo(t)f Fm(\025)g Fo(T)0 3067 y Fu(w)m(e)i(ha)m(v)m(e)g Fo(\036)428 3031 y Fn(t)464 3067 y Fu(\()p Fo(U)11 b Fu(\))k Fm(\\)g Fo(V)50 b Fm(6)p Fu(=)28 b Fm(;)p Fu(.)43 b(W)-8 b(e)30 b(sa)m(y)f(that)h(the)g (\015o)m(w)g(is)g Fo(")p Fu(-)p Fj(top)-5 b(olo)g(gic)g(al)5 b(ly)33 b(mixing)d Fu(if)f(giv)m(en)h(an)m(y)g(t)m(w)m(o)f(non)0 3187 y(empt)m(y)i(op)s(en)h(sets)h Fo(U;)17 b(V)49 b Fm(\032)28 b Fo(X)39 b Fu(whic)m(h)33 b(con)m(tain)f(balls)f(of)h (diameter)f Fo(")p Fu(,)h(there)h(exists)f Fo(T)41 b(>)28 b Fu(0)k(suc)m(h)h(that)0 3306 y(for)g(an)m(y)h Fo(t)27 b Fm(\025)h Fo(T)48 b Fu(w)m(e)33 b(ha)m(v)m(e)h Fo(\036)1050 3270 y Fn(t)1086 3306 y Fu(\()p Fo(U)11 b Fu(\))21 b Fm(\\)i Fo(V)50 b Fm(6)p Fu(=)28 b Fm(;)p Fu(.)120 3468 y(Let)j Fo(X)39 b Fu(b)s(e)32 b(a)f Fo(C)711 3432 y Fk(1)788 3468 y Fu(manifold)f(and)i Fo(\036)1449 3432 y Fn(t)1512 3468 y Fu(:)c Fo(X)35 b Fm(!)27 b Fo(X)39 b Fu(b)s(e)32 b(a)f Fo(C)2228 3432 y Fk(1)2304 3468 y Fu(\015o)m(w.)44 b(Assume)31 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4262 y Fn(s)2582 4298 y Fu(\()p Fo(O)s Fu(\))g(and)i Fo(W)3075 4262 y Fn(u)3127 4298 y Fu(\()p Fo(O)s Fu(\))e(are)h(immersed)0 4418 y Fo(C)78 4382 y Fk(1)156 4418 y Fu(manifolds.)120 4579 y(W)-8 b(e)35 b(require)h(a)g(v)m(ersion)f(of)h(Smale's)f(Homo)s (clinic)f(Theorem)h([S])g(for)h(\015o)m(ws)g(p)s(ossessing)h(a)e (hetero-)0 4699 y(clinic)i(connection.)57 b(Let)37 b Fo(X)45 b Fu(b)s(e)37 b(a)g Fo(C)1427 4663 y Fk(1)1509 4699 y Fu(manifold)f(and)i Fo(\036)2182 4663 y Fn(t)2252 4699 y Fu(:)c Fo(X)41 b Fm(!)34 b Fo(X)45 b Fu(b)s(e)37 b(a)g Fo(C)3004 4663 y Fk(1)3086 4699 y Fu(\015o)m(w.)56 b(Assume)37 b(that)0 4818 y(the)31 b(\015o)m(w)g(p)s(ossesses)i(h)m(yp) s(erb)s(olic)e(closed)g(orbits)g Fo(O)1931 4833 y Fk(1)1976 4818 y Fo(;)17 b(:)g(:)g(:)d(;)j(O)2274 4833 y Fn(m)2379 4818 y Fu(suc)m(h)33 b(that)d(the)h(stable)g(manifold)f(of)h Fo(O)3867 4833 y Fn(i)0 4938 y Fu(in)m(tersects)e(the)g(unstable)g (manifold)e(of)h Fo(O)1588 4953 y Fn(i)p Fk(+1)1751 4938 y Fu(transv)m(ersely)g(at)g Fo(U)2483 4953 y Fn(i)2544 4938 y Fu(for)g(eac)m(h)h Fo(i)f Fu(=)h(1)p Fo(;)17 b(:)g(:)g(:)d(;)j (m)p Fu(.)42 b(Then)29 b(the)0 5058 y(\015o)m(w)39 b(has)f(a)g(lo)s (cally)f(maximal)e(h)m(yp)s(erb)s(olic)j(compact)g(in)m(v)-6 b(arian)m(t)38 b(set)h Fo(K)45 b Fu(\(a)38 b(horsesho)s(e\))h(con)m (taining)0 5177 y Fo(O)76 5192 y Fk(1)149 5177 y Fm([)28 b(\001)17 b(\001)g(\001)27 b([)h Fo(O)558 5192 y Fn(m)676 5177 y Fu(and)42 b Fo(U)946 5192 y Fk(1)1019 5177 y Fm([)29 b(\001)17 b(\001)g(\001)26 b([)i Fo(U)1420 5192 y Fn(m)1496 5177 y Fu(.)71 b(The)42 b(p)s(erio)s(dic)g(orbits)g(are)h(dense)g(in)f Fo(K)49 b Fu(and)43 b(the)f(\015o)m(w)h Fo(\036)3865 5141 y Fn(t)0 5297 y Fu(restricted)34 b(to)f Fo(K)40 b Fu(is)33 b(top)s(ologically)d(mixing.)1929 5639 y Fs(4)p eop %%Page: 5 5 5 4 bop 865 100 a Fw(2.)55 b(Small)36 b(p)s(erturbations)f(to)h(the)g (round)g(metric)120 300 y Fu(In)d(this)f(section)h(w)m(e)g(describ)s(e) h(the)f(t)m(yp)s(es)g(of)g(p)s(erturbation)g(whic)m(h)h(will)d(b)s(e)i (used)h(in)f(Section)g(3)f(to)0 419 y(pro)m(v)m(e)i(Theorems)f(0.1)f (and)i(0.2.)43 b(Before)34 b(doing)f(so,)g(w)m(e)h(mak)m(e)e(some)h (general)g(remarks.)120 559 y(The)39 b(geo)s(desic)h(\015o)m(w)f Fo(\036)999 523 y Fn(t)999 584 y(g)1085 559 y Fu(of)g(a)g(Riemannian)g (metric)f Fo(g)k Fu(on)d Fo(S)2471 523 y Fk(3)2555 559 y Fu(is)g(usually)f(though)m(t)i(of)f(as)g(acting)0 689 y(on)f(the)h(bundle)g Fo(T)723 652 y Fk(1)709 713 y Fn(g)768 689 y Fo(S)835 652 y Fk(3)918 689 y Fu(of)f(v)m(ectors)h(that)f(ha)m(v) m(e)g(unit)g(length)h(with)f(resp)s(ect)h(to)f Fo(g)t Fu(.)58 b(This)38 b(con)m(v)m(en)m(tion)0 808 y(is)f(incon)m(v)m(enien) m(t)i(for)f(us,)h(b)s(ecause)g(it)d(w)m(ould)i(mak)m(e)e(the)i(geo)s (desic)g(\015o)m(ws)g(for)g(di\013eren)m(t)g(metrics)f(act)0 928 y(on)d(di\013eren)m(t)h(bundles.)47 b(Instead)35 b(w)m(e)f(will)f(use)i(radial)e(pro)6 b(jection)34 b(in)g(the)g(\014b)s (ers)h(of)f Fo(T)14 b(S)3370 891 y Fk(3)3448 928 y Fu(to)34 b(iden)m(tify)0 1047 y Fo(T)72 1011 y Fk(1)58 1072 y Fn(g)117 1047 y Fo(S)184 1011 y Fk(3)264 1047 y Fu(with)i(the)h(unit)f (tangen)m(t)g(bundle)h(of)g(the)f(standard)h(round)g(metric)e Fo(g)2938 1062 y Fk(0)2982 1047 y Fu(.)53 b(The)36 b(notation)g Fo(T)3744 1011 y Fk(1)3788 1047 y Fo(S)3855 1011 y Fk(3)0 1176 y Fu(will)42 b(mean)g Fo(T)545 1140 y Fk(1)531 1201 y Fn(g)570 1211 y Fh(0)614 1176 y Fo(S)681 1140 y Fk(3)726 1176 y Fu(.)73 b(It)43 b(will)f(b)s(e)h(con)m(v)m(enien)m(t)i(to)d(use) i(the)g(Sasaki)e(metric)g(and)i(Liouville)e(measure)0 1306 y(\(normalized)29 b(to)g(b)s(e)h(a)g(probabilit)m(y)e(measure\))i (for)g Fo(g)2001 1321 y Fk(0)2075 1306 y Fu(to)f(measure)h(distances)h (and)f(v)m(olume)e(in)i Fo(T)3716 1270 y Fk(1)3761 1306 y Fo(S)3828 1270 y Fk(3)3872 1306 y Fu(.)0 1425 y(W)-8 b(e)36 b(in)m(terpret)g(Theorems)f(0.1)g(and)h(0.2)f(in)g(this)g(w)m(a) m(y)-8 b(.)51 b(It)35 b(is)g(easily)g(seen)h(that)f(these)i(theorems)e (still)0 1545 y(hold)45 b(if)g(the)g(distance)g(and)g(measure)g (de\014ned)i(on)e Fo(T)2100 1509 y Fk(1)2144 1545 y Fo(S)2211 1509 y Fk(3)2301 1545 y Fu(b)m(y)f(the)h(p)s(erturb)s(ed)h(metric)e Fo(g)k Fu(are)d(used)0 1664 y(instead.)120 1805 y(The)38 b(basis)h(of)f(our)g(construction)h(is)f(the)g(follo)m(wing)g(prop)s (osition,)g(whic)m(h)h(allo)m(ws)f(us)h(to)e(p)s(erturb)0 1924 y(the)43 b(round)g(metric)e(on)i Fo(S)1012 1888 y Fk(2)1099 1924 y Fu(so)f(as)h(to)e(mak)m(e)h(giv)m(en)g(orbits)g(of)h (the)f(geo)s(desic)h(\015o)m(w)g(h)m(yp)s(erb)s(olic)f(and)0 2044 y(create)34 b(a)f(hetero)s(clinic)g(orbit)g(connecting)h(them.)0 2285 y Fw(Prop)s(osition)28 b(2.1.)58 b Fj(L)-5 b(et)27 b Fo(\015)1060 2300 y Fk(1)1132 2285 y Fj(and)g Fo(\015)1369 2300 y Fk(2)1441 2285 y Fj(b)-5 b(e)27 b(ge)-5 b(ometric)g(al)5 b(ly)28 b(distinct)e(ge)-5 b(o)g(desics)25 b(in)i Fo(S)3092 2249 y Fk(2)3164 2285 y Fj(with)g(the)h(standar)-5 b(d)0 2404 y(metric.)62 b(We)40 b(c)-5 b(an)41 b(cho)-5 b(ose)40 b(a)h(metric)g Fo(g)1528 2419 y Fk(1)1613 2404 y Fj(on)g Fo(S)1828 2368 y Fk(2)1873 2404 y Fj(,)h(arbitr)-5 b(arily)41 b(close)g(\(in)f(the)h Fo(C)3091 2368 y Fi(1)3217 2404 y Fj(top)-5 b(olo)g(gy\))41 b(to)g(the)0 2524 y(r)-5 b(ound)35 b(metric,)g(such)g(that)g Fo(\015)1109 2539 y Fk(1)1188 2524 y Fj(and)g Fo(\015)1433 2539 y Fk(2)1512 2524 y Fj(ar)-5 b(e)34 b(ge)-5 b(o)g(desics)33 b(for)i Fo(g)2309 2539 y Fk(1)2353 2524 y Fj(,)g(the)f(c)-5 b(orr)g(esp)g (onding)34 b(close)-5 b(d)34 b(orbits)g Fo(O)3855 2539 y Fk(1)0 2643 y Fj(and)c Fo(O)264 2658 y Fk(2)339 2643 y Fj(of)g(the)g(ge)-5 b(o)g(desic)28 b(\015ow)i(for)g Fo(g)1395 2658 y Fk(1)1469 2643 y Fj(ar)-5 b(e)30 b(hyp)-5 b(erb)g(olic,)30 b(and)g Fo(W)2417 2607 y Fn(u)2469 2643 y Fu(\()p Fo(O)2584 2658 y Fk(1)2629 2643 y Fu(\))10 b Fm(\\)g Fo(W)2862 2607 y Fn(s)2905 2643 y Fu(\()p Fo(O)3020 2658 y Fk(2)3064 2643 y Fu(\))28 b Fm(6)p Fu(=)g Fm(;)p Fj(.)43 b(F)-8 b(urthermor)j(e)0 2763 y(this)49 b(p)-5 b(erturb)g(ation)49 b(c)-5 b(an)49 b(b)-5 b(e)49 b(made)g(in)g(an)h (arbitr)-5 b(arily)49 b(smal)5 b(l)50 b(neighb)-5 b(ourho)g(o)g(d)49 b(of)f(one)i(of)f(the)g(two)0 2883 y(interse)-5 b(ction)34 b(p)-5 b(oints)35 b(of)g Fo(\015)1003 2898 y Fk(1)1083 2883 y Fj(and)g Fo(\015)1328 2898 y Fk(2)1373 2883 y Fj(.)0 3124 y(Pr)-5 b(o)g(of.)49 b Fu(Let)30 b Fo(p)f Fu(b)s(e)h(one)f(of)h(the)f(t)m(w)m(o)h(an)m(tip)s(o)s(dal)e(p)s(oin)m (ts)i(in)f(whic)m(h)h(the)g(geo)s(desics)g Fo(\015)3115 3139 y Fk(1)3188 3124 y Fu(and)g Fo(\015)3430 3139 y Fk(2)3503 3124 y Fu(in)m(tersect.)0 3243 y(Let)36 b Fo(v)229 3258 y Fk(1)309 3243 y Fu(and)g Fo(v)553 3258 y Fk(2)634 3243 y Fu(b)s(e)f(the)h(v)m(ectors)g(based)g(at)g Fo(p)f Fu(that)g(b)s(elong)h(to)f Fo(O)2514 3258 y Fk(1)2594 3243 y Fu(and)h Fo(O)2866 3258 y Fk(2)2946 3243 y Fu(resp)s(ectiv)m (ely)-8 b(.)50 b(W)-8 b(e)36 b(shall)0 3363 y(use)c(geo)s(desic)h(p)s (olar)e(co)s(ordinates)g(\(for)h(the)g(round)g(metric\))f(on)g Fo(S)2540 3327 y Fk(2)2616 3363 y Fu(with)h Fo(p)f Fu(as)h(cen)m(ter:) 44 b(let)32 b Fo(r)e Fm(2)e Fu([0)p Fo(;)17 b(\031)t Fu(])0 3482 y(b)s(e)30 b(the)g(radial)f(co)s(ordinate)g(and)i Fo(\022)f Fm(2)e Fo(S)1494 3446 y Fk(1)1568 3482 y Fu(the)i(angular)f (co)s(ordinate.)43 b(W)-8 b(e)30 b(think)f(of)h Fo(S)3224 3446 y Fk(1)3298 3482 y Fu(as)g([0)p Fo(;)17 b Fu(2)p Fo(\031)t Fu(])27 b(with)0 3604 y(its)38 b(endp)s(oin)m(ts)h(iden)m (ti\014ed.)59 b(In)38 b(these)h(co)s(ordinates,)g(the)f(round)h(metric) d(on)i Fo(S)3035 3568 y Fk(2)3118 3604 y Fu(is)g Fo(dr)3323 3568 y Fk(2)3392 3604 y Fu(+)26 b(sin)3617 3562 y Fk(2)3662 3604 y Fo(r)19 b(d\022)3828 3568 y Fk(2)3872 3604 y Fu(.)0 3724 y(Let)29 b Fo(\022)221 3739 y Fk(1)294 3724 y Fu(and)g Fo(\022)530 3739 y Fk(2)603 3724 y Fu(b)s(e)g(the)g(v)-6 b(alues)29 b(of)g(the)g Fo(\022)i Fu(co)s(ordinate)d(on)h(the)g(geo)s (desic)g(ra)m(ys)f(emanating)g(from)g Fo(p)g Fu(that)0 3843 y(are)38 b(tangen)m(t)h(to)f Fo(v)714 3858 y Fk(1)797 3843 y Fu(and)h Fo(v)1044 3858 y Fk(2)1127 3843 y Fu(resp)s(ectiv)m (ely)-8 b(.)59 b(The)38 b(h)m(yp)s(otheses)i(imply)d(that)g Fo(\022)2990 3858 y Fk(1)3073 3843 y Fu(and)i Fo(\022)3319 3858 y Fk(2)3402 3843 y Fu(are)f(distinct)0 3963 y(and)29 b(not)g(an)m(tip)s(o)s(dal)f(to)h(one)g(another.)43 b(W)-8 b(e)29 b(ma)m(y)e(assume)i(that)g(the)g(direction)g(in)g(whic)m(h)h(w)m (e)f(measure)0 4082 y Fo(\022)36 b Fu(and)d(the)h(p)s(osition)f(of)g (the)h(ra)m(y)f Fo(\022)d Fu(=)e(0)33 b(w)m(ere)h(c)m(hosen)h(so)e (that)g(0)27 b Fo(<)i(\022)2667 4097 y Fk(1)2739 4082 y Fo(<)f(\022)2891 4097 y Fk(2)2963 4082 y Fo(<)g(\031)t Fu(.)120 4222 y(Cho)s(ose)33 b(a)g Fo(C)627 4186 y Fi(1)745 4222 y Fu(\015o)m(w)h Fo(\013)1017 4237 y Fn(t)1085 4222 y Fu(on)f Fo(S)1290 4186 y Fk(1)1368 4222 y Fu(suc)m(h)h(that)152 4393 y(\(1\))49 b(the)32 b(only)g(\014xed)g(p)s(oin)m(ts)g(in)g([)p Fo(\022)1444 4408 y Fk(1)1488 4393 y Fo(;)17 b(\022)1580 4408 y Fk(2)1624 4393 y Fu(])32 b(are)g(a)g(h)m(yp)s(erb)s(olic)g (source)h(at)e Fo(\022)2888 4408 y Fk(1)2965 4393 y Fu(and)h(a)g(h)m (yp)s(erb)s(olic)g(sink)329 4512 y(at)h Fo(\022)498 4527 y Fk(2)542 4512 y Fu(;)g(and)152 4632 y(\(2\))49 b(ev)m(ery)33 b(p)s(oin)m(t)g(of)h(the)f(arc)g([)p Fo(\031)26 b Fu(+)c Fo(\022)1560 4647 y Fk(1)1605 4632 y Fo(;)17 b(\031)24 b Fu(+)f Fo(\022)1878 4647 y Fk(2)1923 4632 y Fu(])32 b(\(whic)m(h)i(is)f(an)m(tip)s(o)s(dal)g(to)g([)p Fo(\022)3050 4647 y Fk(1)3094 4632 y Fo(;)17 b(\022)3186 4647 y Fk(2)3230 4632 y Fu(]\))32 b(is)h(\014xed.)120 4802 y(Cho)s(ose)38 b Fo(\032)e Fm(2)h Fu(\(0)p Fo(;)17 b(\031)t Fu(\))37 b(and)i(a)f Fo(C)1298 4766 y Fi(1)1421 4802 y Fu(function)i Fo(\034)49 b Fu(that)39 b(is)f(de\014ned)i(on)f([0)p Fo(;)17 b(\031)t Fu(],)38 b(0)g(on)h([0)p Fo(;)17 b(\032=)p Fu(3],)37 b(nonde-)0 4921 y(creasing)j(on)g([)p Fo(\032=)p Fu(3)p Fo(;)17 b Fu(2)p Fo(\032=)p Fu(3],)37 b(and)j(constan)m(t)g(and) g(p)s(ositiv)m(e)f(on)h([2)p Fo(\032=)p Fu(3)p Fo(;)17 b(\031)t Fu(].)59 b(Let)40 b Fo(f)49 b Fu(:)37 b Fo(S)3248 4885 y Fk(2)3331 4921 y Fm(!)h Fo(S)3536 4885 y Fk(2)3620 4921 y Fu(b)s(e)i(the)0 5041 y(di\013eomorphism)33 b(that)h(maps)f(the) i(p)s(oin)m(t)f(with)g(co)s(ordinates)g(\()p Fo(r)m(;)17 b(\022)s Fu(\))33 b(to)g(the)i(p)s(oin)m(t)f(with)g(co)s(ordinates)0 5160 y(\()p Fo(r)m(;)17 b(\013)190 5178 y Fn(\034)8 b Fk(\()p Fn(r)r Fk(\))340 5160 y Fu(\()p Fo(\022)s Fu(\)\).)56 b(It)37 b(follo)m(ws)g(from)g(\(1\),)h(\(2\))f(and)h(the)g (de\014nition)g(of)f Fo(f)48 b Fu(that)38 b Fo(f)48 b Fu(\014xes)38 b(all)e(p)s(oin)m(ts)i(on)g Fo(\015)3856 5175 y Fk(1)0 5280 y Fu(and)29 b Fo(\015)241 5295 y Fk(2)286 5280 y Fu(.)42 b(The)30 b(circles)f Fo(r)h Fu(=)f(const)g(are)g(mapp)s (ed)g(to)f(themselv)m(es.)43 b(Outside)30 b(the)f(circle)g Fo(r)i Fu(=)d(2)p Fo(\032=)p Fu(3,)g(eac)m(h)0 5400 y(ra)m(y)33 b Fo(\022)d Fu(=)e(const)34 b(is)f(mapp)s(ed)g(to)g(another)h(ra)m(y)e (of)i(this)f(form.)1929 5639 y Fs(5)p eop %%Page: 6 6 6 5 bop 1225 1440 a @beginspecial 45 @llx 486 @lly 319 @urx 758 @ury 1450 @rwi @setspecial %%BeginDocument: pert_met.ps % % Due to bugs in Transcript, the 'PS-Adobe-' stuff is omitted from line 1 % /tgifdict 132 dict def tgifdict begin % % Using a zero value radius for an ellipse or an arc would result % in a non-invertible CTM matrix which causes problem when this % when this PostScript is wrapped inside other routines, such as % the multi.ps package from % ftp.ucc.su.oz.au:/pub/ps_printing/multi. You can overcome such % error by uncommenting the sole line of the procedure below: % /tgif_min_radius { % dup 0.01 lt { pop 0.01 } if } bind def /tgifellipsedict 6 dict def tgifellipsedict /mtrx matrix put /tgifellipse { tgifellipsedict begin /yrad exch def /xrad exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y translate xrad yrad scale 0 0 1 0 360 arc savematrix setmatrix end } def /tgifarrowtipdict 8 dict def tgifarrowtipdict /mtrx matrix put /tgifarrowtip { tgifarrowtipdict begin /dy exch def /dx exch def /h exch def /w exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y translate dy dx atan rotate 0 0 moveto w neg h lineto w neg h neg lineto savematrix setmatrix end } def /tgifarcdict 8 dict def tgifarcdict /mtrx matrix put /tgifarcn { tgifarcdict begin /endangle exch def /startangle exch def /yrad exch def /xrad exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y translate xrad yrad scale 0 0 1 startangle endangle arc savematrix setmatrix end } def /tgifarc { tgifarcdict begin /endangle exch def /startangle exch def /yrad exch def /xrad exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y translate xrad yrad scale 0 0 1 startangle endangle arcn savematrix setmatrix end } def /tgifsetuserscreendict 22 dict def tgifsetuserscreendict begin /tempctm matrix def /temprot matrix def /tempscale matrix def /concatprocs { /proc2 exch cvlit def /proc1 exch cvlit def /newproc proc1 length proc2 length add array def newproc 0 proc1 putinterval newproc proc1 length proc2 putinterval newproc cvx } def /resmatrix matrix def /findresolution { 72 0 resmatrix defaultmatrix dtransform /yres exch def /xres exch def xres dup mul yres dup mul add sqrt } def end /tgifsetuserscreen { tgifsetuserscreendict begin /spotfunction exch def /screenangle exch def /cellsize exch def /m tempctm currentmatrix def /rm screenangle temprot rotate def /sm cellsize dup tempscale scale def sm rm m m concatmatrix m concatmatrix pop 1 0 m dtransform /y1 exch def /x1 exch def /veclength x1 dup mul y1 dup mul add sqrt def /frequency findresolution veclength div def /newscreenangle y1 x1 atan def m 2 get m 1 get mul m 0 get m 3 get mul sub 0 gt {{neg} /spotfunction load concatprocs /spotfunction exch def } if frequency newscreenangle /spotfunction load setscreen end } def /tgifsetpatterndict 18 dict def tgifsetpatterndict begin /bitison { /ybit exch def /xbit exch def /bytevalue bstring ybit bwidth mul xbit 8 idiv add get def /mask 1 7 xbit 8 mod sub bitshift def bytevalue mask and 0 ne } def end /tgifbitpatternspotfunction { tgifsetpatterndict begin /y exch def /x exch def /xindex x 1 add 2 div bpside mul cvi def /yindex y 1 add 2 div bpside mul cvi def xindex yindex bitison { /onbits onbits 1 add def 1 } { /offbits offbits 1 add def 0 } ifelse end } def /tgifsetpattern { tgifsetpatterndict begin /cellsz exch def /angle exch def /bwidth exch def /bpside exch def /bstring exch def /onbits 0 def 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b(cir)-5 b(cles)45 b Fo(\033)3855 1006 y Fk(1)0 1110 y Fj(and)50 b Fo(\033)265 1125 y Fk(2)309 1110 y Fj(.)88 b(L)-5 b(et)49 b Fo(\015)665 1125 y Fk(1)759 1110 y Fj(and)h Fo(\015)1019 1125 y Fk(2)1113 1110 y Fj(b)-5 b(e)49 b(ge)-5 b(o)g(desics)48 b(in)h Fo(S)1889 1074 y Fk(3)1983 1110 y Fj(obtaine)-5 b(d)49 b(by)h(cho)-5 b(osing)48 b(dir)-5 b(e)g(ctions)48 b(for)h Fo(\033)3648 1125 y Fk(1)3742 1110 y Fj(and)0 1230 y Fo(\033)57 1245 y Fk(2)144 1230 y Fj(r)-5 b(esp)g(e)g(ctively.)66 b(Then)43 b(the)g(change)f(of)g (metric)h Fo(g)1999 1245 y Fk(1)2086 1230 y Fj(on)g Fu(\006)f Fj(describ)-5 b(e)g(d)42 b(in)g(Pr)-5 b(op)g(osition)41 b(2.1)h(c)-5 b(an)42 b(b)-5 b(e)0 1349 y(achieve)g(d)41 b(by)h(a)h(change)e(of)h(metric)g(in)g(an)h(arbitr)-5 b(arily)42 b(smal)5 b(l)43 b(b)-5 b(al)5 b(l)43 b Fo(B)k Fj(in)42 b Fo(S)2954 1313 y Fk(3)3041 1349 y Fj(ar)-5 b(ound)42 b(one)h(the)f(two)0 1469 y(p)-5 b(oints)36 b(in)h Fu(\006)23 b Fm(\\)h Fu(\006)675 1484 y Fk(1)742 1469 y Fm(\\)g Fu(\006)904 1484 y 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Fo(q)k Fu(to)c(\006.)44 b(These)34 b(co)s(ordinates)g(are)f(illustrated)g(in)g(Figure)g(3.)1438 4065 y @beginspecial 76 @llx 519 @lly 245 @urx 733 @ury 1023 @rwi @setspecial %%BeginDocument: s_3.ps % % Due to bugs in Transcript, the 'PS-Adobe-' stuff is omitted from line 1 % /tgifdict 132 dict def tgifdict begin % % Using a zero value radius for an ellipse or an arc would result % in a non-invertible CTM matrix which causes problem when this % when this PostScript is wrapped inside other routines, such as % the multi.ps package from % ftp.ucc.su.oz.au:/pub/ps_printing/multi. You can overcome such % error by uncommenting the sole line of the procedure below: % /tgif_min_radius { % dup 0.01 lt { pop 0.01 } if } bind def /tgifellipsedict 6 dict def tgifellipsedict /mtrx matrix put /tgifellipse { tgifellipsedict begin /yrad exch def /xrad exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y translate xrad yrad scale 0 0 1 0 360 arc savematrix setmatrix end } def /tgifarrowtipdict 8 dict def tgifarrowtipdict /mtrx matrix put /tgifarrowtip { tgifarrowtipdict begin /dy exch def /dx exch def /h exch def /w exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y translate dy dx atan rotate 0 0 moveto w neg h lineto w neg h neg lineto savematrix setmatrix end } def /tgifarcdict 8 dict def tgifarcdict /mtrx matrix put /tgifarcn { tgifarcdict begin /endangle exch def /startangle exch def /yrad exch def /xrad exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y translate xrad yrad scale 0 0 1 startangle endangle arc savematrix setmatrix end } def /tgifarc { tgifarcdict begin /endangle exch def /startangle exch def /yrad exch def /xrad exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y translate xrad yrad scale 0 0 1 startangle endangle arcn savematrix setmatrix end } def /tgifsetuserscreendict 22 dict def tgifsetuserscreendict begin /tempctm matrix def /temprot matrix def /tempscale matrix def /concatprocs { /proc2 exch cvlit def /proc1 exch cvlit def /newproc proc1 length proc2 length add array def newproc 0 proc1 putinterval newproc proc1 length proc2 putinterval newproc cvx } def /resmatrix matrix def /findresolution { 72 0 resmatrix defaultmatrix dtransform /yres exch def /xres exch def xres dup mul yres dup mul add sqrt } def end /tgifsetuserscreen { tgifsetuserscreendict begin /spotfunction exch def /screenangle exch def /cellsize exch def /m tempctm currentmatrix def /rm screenangle temprot rotate def /sm cellsize dup tempscale scale def sm rm m m concatmatrix m concatmatrix pop 1 0 m dtransform /y1 exch def /x1 exch def /veclength x1 dup mul y1 dup mul add sqrt def /frequency findresolution veclength div def /newscreenangle y1 x1 atan def m 2 get m 1 get mul m 0 get m 3 get mul sub 0 gt {{neg} /spotfunction load concatprocs /spotfunction exch def } if frequency newscreenangle /spotfunction load setscreen end } def /tgifsetpatterndict 18 dict def tgifsetpatterndict begin /bitison { /ybit exch def /xbit exch def /bytevalue bstring ybit bwidth mul xbit 8 idiv add get def /mask 1 7 xbit 8 mod sub bitshift def bytevalue mask and 0 ne } def end /tgifbitpatternspotfunction { tgifsetpatterndict begin /y exch def /x exch def /xindex x 1 add 2 div bpside mul cvi def /yindex y 1 add 2 div bpside mul cvi def xindex yindex bitison { /onbits onbits 1 add def 1 } { /offbits offbits 1 add def 0 } ifelse end } def /tgifsetpattern { tgifsetpatterndict begin /cellsz exch def /angle exch def /bwidth exch def /bpside exch def /bstring exch def /onbits 0 def /offbits 0 def cellsz angle /tgifbitpatternspotfunction load tgifsetuserscreen {} settransfer offbits offbits onbits add div setgray end } def /tgifxpmdict 4 dict def /tgifbwpicstr 1 string def /tgifcolorpicstr 3 string def /tgifsetpixels { tgifxpmdict begin /pixels exch def end } def /tgifsetpix { tgifxpmdict begin pixels 3 1 roll putinterval end } def /tgifbwspot { tgifxpmdict begin /index exch def tgifbwpicstr 0 pixels index 3 mul 3 getinterval aload pop 255 mul .114 mul exch 255 mul .587 mul add exch 255 mul .299 mul add cvi put tgifbwpicstr end } def /tgifcolorspot { tgifxpmdict begin /index exch def pixels index 3 mul 3 getinterval aload pop 255 mul cvi tgifcolorpicstr 2 3 -1 roll put 255 mul cvi tgifcolorpicstr 1 3 -1 roll put 255 mul cvi tgifcolorpicstr 0 3 -1 roll put tgifcolorpicstr end } def /tgifnewcolorspot { tgifxpmdict begin /index exch def pixels index 3 mul 3 getinterval aload pop setrgbcolor end } def /tgifcolordict 4 dict def /colorimage where { pop } { /colorimage { tgifcolordict begin pop pop pop pop pop /ih exch def /iw exch def /x 0 def /y 0 def 1 1 ih { pop 1 1 iw { pop currentfile tgifbwpicstr readhexstring pop 0 get tgifnewcolorspot x y moveto 1 0 rlineto 0 1 rlineto -1 0 rlineto closepath fill /x x 1 add def } for /y y 1 add def /x 0 def } for end } def } ifelse /tgifpatdict 10 dict def /tgifpatbyte { currentdict /retstr get exch pat i cellsz mod get put } def /tgifpatproc { 0 1 widthlim {tgifpatbyte} for retstr /i i 1 add def } def /tgifpatfill { tgifpatdict begin /h exch def /w exch def /lty exch def /ltx exch def /cellsz exch def /pat exch def /widthlim w cellsz div cvi 1 sub def /retstr widthlim 1 add string def /i 0 def ltx lty translate w h true [1 0 0 1 0 0] {tgifpatproc} imagemask ltx neg lty neg translate end } def /pat1 def /pat2 <0000000000000000> def /pat3 <8000000008000000> def /pat4 <8800000022000000> def /pat5 <8800220088002200> def /pat6 <8822882288228822> def /pat7 def /pat8 <77dd77dd77dd77dd> def /pat9 <77ffddff77ffddff> def /pat10 <77ffffff77ffffff> def /pat11 <7fffffff7fffffff> def /pat12 <8040200002040800> def /pat13 <40a00000040a0000> def /pat14 def /pat15 def /pat16 def /pat17 <038448300c020101> def /pat18 <081c22c180010204> def /pat19 <8080413e080814e3> def /pat20 <8040201008040201> def /pat21 <8844221188442211> def /pat22 <77bbddee77bbddee> def /pat23 def /pat24 <7fbfdfeff7fbfdfe> def /pat25 <3e1f8fc7e3f1f87c> def /pat26 <0102040810204080> def /pat27 <1122448811224488> def /pat28 def /pat29 <83070e1c3870e0c1> def /pat30 def /pat31 <7cf8f1e3c78f1f3e> def /tgifcentertext { dup stringwidth pop 2 div neg 0 rmoveto } def /tgifrighttext { dup stringwidth pop neg 0 rmoveto } def /tgifreencsmalldict 12 dict def /tgifReEncodeSmall { tgifreencsmalldict begin /newcodesandnames exch def /newfontname exch def /basefontname exch def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict { exch dup /FID ne { dup /Encoding eq { exch dup length array copy newfont 3 1 roll put } { exch newfont 3 1 roll put } ifelse } { pop pop } ifelse } forall newfont /FontName newfontname put newcodesandnames aload pop newcodesandnames length 2 idiv { newfont /Encoding get 3 1 roll put} repeat newfontname newfont definefont pop end } def /tgifgray { 8 1 0 72 300 32 div div tgifsetpattern } bind def /tgifboxdict 6 dict def /tgifboxstroke { tgifboxdict begin /pat def /w def /y2 exch def /x2 exch def /y1 exch def /x1 exch def 1.415 setmiterlimit w 1 eq { w setlinewidth } if pat pat1 ne pat pat2 ne and { gsave pat tgifgray } if newpath x1 y1 moveto x2 y1 lineto x2 y2 lineto x1 y2 lineto closepath pat pat2 eq { 1 setgray stroke 0 setgray } { stroke } ifelse pat pat1 ne pat pat2 ne and { grestore } if w 1 eq { 1 setlinewidth } if 1 setmiterlimit end } def /tgifboxfill { tgifboxdict begin /pat def /y2 exch def /x2 exch def /y1 exch def /x1 exch def pat pat1 ne pat pat2 ne and { gsave pat tgifgray } if newpath x1 y1 moveto x2 y1 lineto x2 y2 lineto x1 y2 lineto closepath pat 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setlinewidth newpath 229 323 9 9 -116 18 tgifarcn stroke 1 setlinewidth grestore % TEXT 0 setgray /Helvetica-Bold findfont [17 0 0 -17 0 0] makefont setfont gsave 147 123 moveto (great circle orthogonal) show 147 143 moveto ( to) show grestore % TEXT 0 setgray /Symbol findfont [20 0 0 -20 0 0] makefont setfont gsave 220 143 moveto (S) show grestore grestore tgifsavedpage restore end showpage %MatchingCreationDate: Mon Jul 20 12:06:41 1998 %%EndDocument @endspecial 820 4431 a Fv(Figure)k(3.)159 b Fu(\()p Fo(\022)s(;)17 b(r)m(;)g(s)p Fu(\))36 b Fv(Coordina)-7 b(te)38 b(System)g(on)g Fo(S)3036 4395 y Fk(3)120 4793 y Fu(Cho)s(ose)30 b Fo(\032)515 4808 y Fk(0)586 4793 y Fo(>)f Fu(0)g(su\016cien)m(tly)i(small)d(suc)m(h)j(that)f(the)g(n)m (um)m(b)s(er)g Fo(\032)f Fu(c)m(hosen)j(during)e(the)g(construction)0 4913 y(of)36 b(the)f(metric)g Fo(g)650 4928 y Fk(1)729 4913 y Fu(on)h(\006)f(in)g(Prop)s(ositon)g(2.1)g(satis\014es)h Fo(\032)31 b(<)g(\032)2381 4928 y Fk(0)2426 4913 y Fu(.)50 b(The)36 b(metric)e Fo(g)3071 4928 y Fk(1)3151 4913 y Fu(on)h(\006)h(di\013ers)g(from)0 5032 y(the)e(round)g(metric)e(only)h (on)g(\006)22 b Fm(\\)h Fo(B)5 b Fu(\()p Fo(p;)17 b(\032)1577 5047 y Fk(0)1619 5032 y Fu(\).)120 5161 y(Let)50 b Fo(\014)5 b Fu(\()p Fo(s)p Fu(\))49 b(b)s(e)h(a)g(bump)g(function)h(suc)m(h)g (that)e Fo(\014)5 b Fu(\()p Fo(s)p Fu(\))56 b(=)f(1)50 b(if)g Fm(j)p Fo(s)p Fm(j)k(\024)i Fo(\032)2905 5176 y Fk(0)2949 5161 y Fo(=)p Fu(3,)d(0)j Fm(\024)f Fo(\014)5 b Fu(\()p Fo(s)p Fu(\))55 b Fm(\024)h Fu(1)50 b(if)0 5280 y Fo(\032)52 5295 y Fk(0)96 5280 y Fo(=)p Fu(3)28 b Fm(\024)g Fo(s)f Fm(\024)h Fu(2)p Fo(\032)610 5295 y Fk(0)654 5280 y Fo(=)p Fu(3)c(and)h Fo(\014)5 b Fu(\()p Fo(s)p Fu(\))27 b(=)h(0)c(if)g Fo(s)k Fm(\025)g Fu(2)p Fo(\032)1719 5295 y Fk(0)1763 5280 y Fo(=)p Fu(3.)40 b(Let)25 b Fo(F)38 b Fu(b)s(e)24 b(the)h(di\013eomorphism)e(of)h Fo(B)32 b Fu(=)d Fo(B)5 b Fu(\()p Fo(p;)17 b Fu(2)p Fo(\032)3819 5295 y Fk(0)3861 5280 y Fu(\))0 5400 y(that)24 b(maps)g(the)g(p)s(oin)m (t)g(with)g(co)s(ordinates)h(\()p Fo(r)m(;)17 b(\022)s(;)g(s)p Fu(\))k(to)j(the)h(p)s(oin)m(t)f(with)g(co)s(ordinates)g(\()p Fo(r)m(;)17 b(\013)3394 5418 y Fn(\014)t Fk(\()p Fn(s)p Fk(\))p Fn(\034)8 b Fk(\()p Fn(r)r Fk(\))3693 5400 y Fo(\022)s(;)17 b(s)p Fu(\),)1929 5639 y Fs(7)p eop %%Page: 8 8 8 7 bop 0 100 a Fu(where)32 b Fo(\013)350 115 y Fn(t)415 100 y Fu(is)f(the)g(\015o)m(w)h(used)g(in)f(the)g(construction)g(of)g Fo(f)42 b Fu(in)31 b(the)g(pro)s(of)g(of)g(Prop)s(osition)f(2.1.)42 b(Observ)m(e)0 219 y(that)33 b Fo(F)47 b Fu(has)34 b(prop)s(erties)f (analogous)g(to)g(those)h(of)f Fo(f)11 b Fu(:)229 369 y Fm(\017)50 b Fu(The)34 b(lev)m(el)e(surfaces)j(of)f(the)g(co)s (ordinates)f Fo(r)j Fu(and)e Fo(s)e Fu(are)i(mapp)s(ed)f(in)m(to)g (themselv)m(es;)229 488 y Fm(\017)50 b Fu(Outside)30 b(the)g(region)f(where)h Fo(r)h Fm(\024)d Fu(2)p Fo(\032=)p Fu(3,)g(one)i(lev)m(el)f(surface)i(of)e(the)h(co)s(ordinate)f Fo(\022)j Fu(is)d(mapp)s(ed)329 608 y(to)k(another)g(lev)m(el)g (surface)i(of)e Fo(\022)s Fu(.)0 757 y(F)-8 b(urthermore,)229 907 y Fm(\017)50 b Fu(The)34 b(function)g Fo(F)47 b Fu(is)33 b(the)h(iden)m(tit)m(y)f(outside)g(the)h(region)f(where)h Fm(j)p Fo(s)p Fm(j)27 b(\024)h Fo(\032)3064 922 y Fk(0)3108 907 y Fu(.)120 1058 y(In)33 b(the)h(\()p Fo(r)m(;)17 b(\022)s(;)g(s)p Fu(\))30 b(co)s(ordinates,)j(the)h(round)g(metric)e (on)h Fo(S)2285 1022 y Fk(3)2363 1058 y Fu(is)g(cos)2596 1022 y Fk(2)2641 1058 y Fo(s)17 b Fu(\()p Fo(dr)2844 1022 y Fk(2)2909 1058 y Fu(+)23 b(sin)3131 1017 y Fk(2)3176 1058 y Fo(r)c(d\022)3342 1022 y Fk(2)3386 1058 y Fu(\))j(+)g Fo(ds)3645 1022 y Fk(2)3690 1058 y Fu(.)44 b(W)-8 b(e)0 1178 y(no)m(w)36 b(de\014ne)h(a)e(new)h(metric)e Fo(g)1152 1193 y Fk(2)1232 1178 y Fu(on)i Fo(B)j Fu(in)d Fo(S)1674 1142 y Fk(3)1754 1178 y Fu(whic)m(h)g(will)e(coincide)i(with)g(the)f (round)i(metric)d(on)i(the)0 1298 y(complemen)m(t)j(of)h Fo(B)5 b Fu(.)63 b(In)40 b Fo(B)k Fu(w)m(e)d(decree)g(that)e(the)i(co)s (ordinate)e(v)m(ector)h(\014elds)h Fo(@)6 b(=@)g(\022)41 b Fu(and)f Fo(@)6 b(=@)g(s)38 b Fu(will)0 1417 y(ha)m(v)m(e)h(the)g (same)f(length)h(as)g(in)g(the)g(round)h(metric)d(and)j(will)d(still)h (b)s(e)h(orthogonal,)g(while)g(the)g(unit)0 1537 y(v)m(ector)31 b(\014eld)g(orthogonal)f(to)g(b)s(oth)h Fo(@)6 b(=@)g(\022)32 b Fu(and)f Fo(@)6 b(=@)g(s)30 b Fu(will)f(no)m(w)i(b)s(e)g(the)h (pushforw)m(ard)g(b)m(y)f Fo(F)45 b Fu(of)31 b Fo(@)6 b(=@)g(r)s Fu(.)0 1656 y(It)35 b(follo)m(ws)h(from)f(the)h(ab)s(o)m(v)m (e)f(prop)s(erties)i(of)f Fo(F)49 b Fu(that)35 b(the)h(new)g(metric)f (agrees)h(with)g(the)g(old)f(metric)0 1776 y(except)j(where)g Fo(r)f Fm(\024)e Fu(2)p Fo(\032=)p Fu(3)i(or)g Fm(j)p Fo(s)p Fm(j)c(\024)i Fo(\032)1468 1791 y Fk(0)1512 1776 y Fu(.)57 b(Near)37 b Fo(p)g Fu(\(where)h Fm(j)p Fo(s)p Fm(j)c Fo(<)h(\032)2561 1791 y Fk(0)2605 1776 y Fo(=)p Fu(3\))i(this)g(new)h(metric)e(tak)m(es)i(the)0 1895 y(form)858 2015 y(cos)992 1974 y Fk(2)1037 2015 y Fo(s)17 b Fu(\()p Fo(a)p Fu(\()p Fo(r)m(;)g(\022)s Fu(\))g Fo(dr)1525 1974 y Fk(2)1588 2015 y Fu(+)22 b(sin)1810 1973 y Fk(2)1854 2015 y Fo(r)e(d\022)2021 1974 y Fk(2)2087 2015 y Fu(+)j(2)p Fo(b)p Fu(\()p Fo(r)m(;)17 b(\022)s Fu(\))g Fo(dr)s(d\022)s Fu(\))h(+)23 b Fo(ds)2970 1974 y Fk(2)3014 2015 y Fo(;)731 b Fu(\(1\))0 2205 y(where)34 b(the)f(functions)h Fo(a)p Fu(\()p Fo(r)m(;)17 b(\022)s Fu(\))32 b(and)h Fo(b)p Fu(\()p Fo(r)m(;)17 b(\022)s Fu(\))31 b(are)i(functions)i(in)m(tro)s (duced)f(in)f(the)g(pro)s(of)g(of)h(Prop)s(ositon)0 2324 y(2.1,)h(whic)m(h)i(giv)m(e)e(the)h(metric)e Fo(g)1220 2339 y Fk(1)1300 2324 y Fu(in)i(the)g(\()p Fo(r)m(;)17 b(\022)s Fu(\))34 b(co)s(ordinates.)51 b(It)36 b(is)f(ob)m(vious)h (that)f(w)m(e)h(can)g(mak)m(e)f Fo(g)3856 2339 y Fk(2)0 2444 y Fu(as)e(close)h(to)f(the)g(round)h(metric)f(as)g(desired)h(b)m (y)g(c)m(ho)s(osing)f Fo(\032)2308 2459 y Fk(0)2386 2444 y Fu(su\016cien)m(tly)g(small.)120 2563 y(It)g(is)h(easily)f(seen)i (that)f Fo(F)47 b Fu(maps)34 b(\006)22 b Fm(\\)h Fo(B)39 b Fu(in)m(to)33 b(itself)h(and)h(maps)e(\006)2681 2578 y Fn(j)2746 2563 y Fm(\\)23 b Fo(B)38 b Fu(in)m(to)c(itself)g(for)g Fo(j)g Fu(=)29 b(1)p Fo(;)17 b Fu(2.)0 2683 y(Moreo)m(v)m(er)33 b Fo(F)46 b Fu(\014xes)33 b(eac)m(h)h(p)s(oin)m(t)e(of)h(\006)1439 2698 y Fn(j)1501 2683 y Fm(\\)21 b Fo(B)37 b Fu(for)c Fo(j)g Fu(=)c(1)p Fo(;)17 b Fu(2,)31 b(and)i(the)f(restriction)h(of)g Fo(F)46 b Fu(to)32 b(\006)20 b Fm(\\)h Fo(B)37 b Fu(is)c Fo(f)11 b Fu(.)120 2803 y(The)25 b(follo)m(wing)f(lemma)f(will)h(b)s(e) h(useful)h(for)f(sho)m(wing)g(that)g(\006)5 b Fm(\\)g Fo(B)30 b Fu(and)25 b(\006)2859 2818 y Fn(i)2898 2803 y Fm(\\)5 b Fo(B)30 b Fu(are)25 b(totally)d(geo)s(desic)0 2922 y(surfaces.)0 3102 y Fw(Lemma)47 b(2.3.)57 b Fj(L)-5 b(et)43 b Fo(x)886 3065 y Fk(1)931 3102 y Fo(;)17 b(:)g(:)g(:)d(;)j(x) 1210 3065 y Fn(n)1306 3102 y Fj(b)-5 b(e)42 b(lo)-5 b(c)g(al)43 b(c)-5 b(o)g(or)g(dinates)41 b(on)i(a)f(R)m(iemannian)g(manifold)h Fu(\()p Fo(M)5 b(;)17 b(g)t Fu(\))40 b Fj(and)0 3221 y(let)h Fo(H)49 b Fj(b)-5 b(e)40 b(the)h(hyp)-5 b(ersurfac)g(e)40 b(de\014ne)-5 b(d)40 b(by)h Fo(x)1686 3185 y Fn(\013)1738 3195 y Fh(0)1818 3221 y Fu(=)c Fo(c)p Fj(,)42 b(wher)-5 b(e)40 b Fo(c)h Fj(is)f(a)h(c)-5 b(onstant.)60 b(Supp)-5 b(ose)41 b(that)f(at)h(al)5 b(l)0 3341 y(p)-5 b(oints)38 b(in)g Fo(H)47 b Fj(we)39 b(have)f Fo(g)982 3356 y Fn(\013)1034 3366 y Fh(0)1072 3356 y Fn(\014)1159 3341 y Fu(=)c(0)39 b Fj(and)f Fo(@)6 b(g)1662 3356 y Fn(\014)t(\015)1761 3341 y Fo(=@)g(x)1927 3305 y Fn(\013)1979 3315 y Fh(0)2055 3341 y Fu(=)34 b(0)39 b Fj(unless)g Fo(\014)g Fu(=)34 b Fo(\013)2826 3356 y Fk(0)2909 3341 y Fj(or)39 b Fo(\015)f Fu(=)c Fo(\013)3306 3356 y Fk(0)3350 3341 y Fj(.)55 b(Then)39 b Fo(H)47 b Fj(is)0 3460 y(total)5 b(ly)37 b(ge)-5 b(o)g(desic.)0 3640 y(Pr)g(o)g(of.)49 b Fu(This)37 b(is)g(easily)g(pro)m(v)m(ed)g(b)m (y)g(computing)g(Christo\013el)f(sym)m(b)s(ols.)55 b(W)-8 b(e)37 b(sk)m(etc)m(h)h(an)f(alternativ)m(e)0 3759 y(pro)s(of.)65 b(Consider)41 b(a)f(curv)m(e)i Fo(\033)h Fu(that)d(joins)h(t)m(w)m(o)f (p)s(oin)m(ts)h Fo(p)f Fu(and)h Fo(q)j Fu(of)d Fo(H)48 b Fu(and)41 b(is)f(a)g(geo)s(desic)h(for)g(the)0 3879 y(induced)j(metric)d(on)h Fo(H)8 b Fu(.)70 b(W)-8 b(e)42 b(need)h(to)f(sho)m(w)h(that)e(if)h(w)m(e)h(v)-6 b(ary)41 b Fo(\033)t Fu(,)i(k)m(eeping)g(its)f(endp)s(oin)m(ts)h(\014xed,)0 3998 y(then)30 b(the)f(energy)h(in)m(tegral)e(is)h(constan)m(t)h(to)e (\014rst)i(order.)43 b(Since)30 b Fo(\033)j Fu(is)c(a)f(geo)s(desic)i (in)f Fo(H)8 b Fu(,)30 b(all)e(v)-6 b(ariations)0 4118 y(tangen)m(t)23 b(to)g Fo(H)31 b Fu(ha)m(v)m(e)24 b(this)f(prop)s(ert)m (y)-8 b(.)40 b(Th)m(us)24 b(it)f(su\016ces)h(to)f(consider)h(a)f(v)-6 b(ariation)22 b(in)h(the)g Fo(x)3371 4082 y Fn(\013)3423 4092 y Fh(0)3490 4118 y Fu(direction.)0 4238 y(Supp)s(ose)32 b Fo( )t Fu(\()p Fo(u;)17 b(t)p Fu(\))29 b(is)h(suc)m(h)i(a)f(v)-6 b(ariation.)42 b(Let)31 b Fo(v)1786 4201 y Fn(\013)1842 4238 y Fu(\()p Fo(u;)17 b(t)p Fu(\))30 b(b)s(e)g(the)h Fo(\013)p Fu('th)g(comp)s(onen)m(t)f(of)h Fo(@)6 b( )t(=@)g(t)p Fu(\()p Fo(u;)17 b(t)p Fu(\))27 b(and)0 4357 y Fo(g)48 4372 y Fn(\013\014)153 4357 y Fu(\()p Fo(u;)17 b(t)p Fu(\))32 b(the)i(metric)e(at)h Fo( )t Fu(\()p Fo(u;)17 b(t)p Fu(\).)42 b(When)34 b Fo(u)28 b Fu(=)g(0)33 b(w)m(e)h(ha)m(v)m (e:)152 4507 y(\(1\))49 b Fo(v)381 4470 y Fn(\013)433 4480 y Fh(0)504 4507 y Fu(=)28 b(0;)152 4626 y(\(2\))49 b Fo(g)377 4641 y Fn(\013)429 4651 y Fh(0)467 4641 y Fn(\014)548 4626 y Fu(=)29 b(0)k(unless)h Fo(\014)f Fu(=)28 b Fo(\013)1289 4641 y Fk(0)1334 4626 y Fu(;)152 4746 y(\(3\))49 b Fo(dv)429 4761 y Fn(\013)486 4746 y Fo(=du)27 b Fu(=)h(0)33 b(unless)i Fo(\013)27 b Fu(=)i Fo(\013)1416 4761 y Fk(0)1460 4746 y Fu(;)152 4865 y(\(4\))49 b Fo(dg)429 4880 y Fn(\013\014)534 4865 y Fo(=du)27 b Fu(=)h(0)33 b(unless)i Fo(\013)27 b Fu(=)i Fo(\013)1464 4880 y Fk(0)1541 4865 y Fu(or)k Fo(\014)h Fu(=)28 b Fo(\013)1922 4880 y Fk(0)1966 4865 y Fu(.)120 5015 y(Hence)309 5229 y Fo(d)p 285 5274 99 4 v 285 5365 a(ds)396 5296 y Fm(k)p Fo(@)6 b( )t(=@)g(t)p Fm(k)769 5255 y Fk(2)839 5296 y Fu(=)979 5229 y Fo(d)p 956 5274 V 956 5365 a(ds)1083 5202 y Ff(X)1093 5416 y Fn(\013;\014)1244 5296 y Fo(g)1292 5311 y Fn(\013\014)1397 5296 y Fo(v)1449 5255 y Fn(\013)1505 5296 y Fo(v)1557 5255 y Fn(\014)1639 5296 y Fu(=)1744 5202 y Ff(X)1754 5416 y Fn(\013;\014)1904 5156 y Ff(\032)1991 5229 y Fo(dg)2091 5244 y Fn(\013\014)p 1991 5274 205 4 v 2039 5365 a Fo(du)2208 5296 y(v)2260 5255 y Fn(\013)2317 5296 y Fo(v)2369 5255 y Fn(\014)2444 5296 y Fu(+)23 b Fo(g)2592 5311 y Fn(\013\014)2709 5229 y Fo(dv)2813 5193 y Fn(\013)p 2709 5274 161 4 v 2735 5365 a Fo(du)2881 5296 y(v)2933 5255 y Fn(\014)3009 5296 y Fu(+)g Fo(g)3157 5311 y Fn(\013\014)3262 5296 y Fo(v)3314 5255 y Fn(\013)3382 5229 y Fo(dv)3486 5193 y Fn(\014)p 3382 5274 158 4 v 3407 5365 a Fo(du)3552 5156 y Ff(\033)1929 5639 y Fs(8)p eop %%Page: 9 9 9 8 bop 0 100 a Fu(v)-6 b(anishes)35 b(when)g Fo(u)30 b Fu(=)f(0.)47 b(In)m(tegrating)34 b(with)g(resp)s(ect)h(to)f Fo(t)g Fu(no)m(w)g(sho)m(ws)h(that)f(deriv)-6 b(ativ)m(e)34 b(at)g Fo(u)29 b Fu(=)h(0)k(of)0 219 y(the)g(energy)f(in)m(tegral)g(is) g(0.)2792 b Fg(\004)120 457 y Fu(W)-8 b(e)35 b(no)m(w)h(sho)m(w)g(that) f(\006)23 b Fm(\\)h Fo(B)40 b Fu(is)35 b(totally)e(geo)s(desic.)51 b(This)36 b(set)f(is)g(the)h(lev)m(el)f(surface)i Fo(s)31 b Fu(=)g(0.)50 b(Since)0 576 y(the)30 b(pushforw)m(ard)i(of)e Fo(@)6 b(=@)g(r)32 b Fu(b)m(y)e Fo(F)43 b Fu(is)30 b(a)g(linear)g(com)m (bination)f(of)h Fo(@)6 b(=@)g(r)31 b Fu(and)g Fo(@)6 b(=@)g(\022)s Fu(,)28 b(it)h(is)h(ob)m(vious)g(that)0 696 y Fo(@)6 b(=@)g(s)36 b Fu(is)i(orthogonal)g(to)f(b)s(oth)h Fo(@)6 b(=@)g(r)39 b Fu(and)g Fo(@)6 b(=@)g(\022)39 b Fu(in)f(the)h(metric)e Fo(g)t Fu(.)57 b(It)38 b(follo)m(ws)g (immediately)d(from)0 815 y(the)f(explicit)e(form)h(\(1\))f(of)i(the)f (metric)f Fo(g)1562 830 y Fk(2)1640 815 y Fu(that)h(on)g(\006)g(w)m(e)h (ha)m(v)m(e)g(that)664 1035 y Fo(d)p 641 1080 99 4 v 641 1171 a(ds)751 1103 y(g)t Fu(\()p Fo(@)6 b(=@)g(r)m(;)17 b(@)6 b(=@)g(r)s Fu(\))23 b(=)1515 1035 y Fo(d)p 1492 1080 V 1492 1171 a(ds)1603 1103 y(g)t Fu(\()p Fo(@)6 b(=@)g(\022)s(;)17 b(@)6 b(=@)g(\022)s Fu(\))21 b(=)2376 1035 y Fo(d)p 2353 1080 V 2353 1171 a(ds)2463 1103 y(g)t Fu(\()p Fo(@)6 b(=@)g(r)m(;)17 b(@)6 b(=@)g(\022)s Fu(\))23 b(=)28 b(0)p Fo(:)120 1366 y Fu(Finally)-8 b(,)33 b(consider)j(\006)942 1381 y Fk(1)1010 1366 y Fm(\\)24 b Fo(B)39 b Fu(and)d(\006)1483 1381 y Fk(2)1551 1366 y Fm(\\)24 b Fo(B)5 b Fu(.)48 b(Aw)m(a)m(y)35 b(from)f Fo(p)p Fu(,)h(these)h(are)f(formed)g(b)m(y)h(lev)m(el)e (surfaces)0 1486 y(of)39 b(the)g(co)s(ordinate)g Fo(\022)s Fu(,)h(namely)d Fo(\022)j Fu(=)d Fo(\022)1509 1501 y Fk(1)1554 1486 y Fu(,)j Fo(\022)f Fu(=)f Fo(\022)1870 1501 y Fk(1)1940 1486 y Fu(+)27 b Fo(\031)t Fu(,)39 b Fo(\022)g Fu(=)f Fo(\022)2420 1501 y Fk(2)2503 1486 y Fu(and)i Fo(\022)f Fu(=)f Fo(\022)2951 1501 y Fk(2)3021 1486 y Fu(+)26 b Fo(\031)t Fu(.)61 b(In)39 b(order)g(to)g(b)s(e)0 1605 y(able)33 b(to)f(apply)h(Lemma)e(2.3,)h(w)m(e)h(need)h(to)f(v)m (erify)f(that)h Fo(@)6 b(=@)g(\022)34 b Fu(is)f(orthogonal)f(to)g Fo(@)6 b(=@)g(r)34 b Fu(and)g Fo(@)6 b(=@)g(s)31 b Fu(on)0 1725 y(\(\006)111 1740 y Fk(1)178 1725 y Fm([)22 b Fu(\006)338 1740 y Fk(2)383 1725 y Fu(\))g Fm(\\)g Fo(B)38 b Fu(and)679 1945 y Fo(d)p 654 1989 102 4 v 654 2081 a(d\022)768 2012 y(g)t Fu(\()p Fo(@)6 b(=@)g(r)m(;)17 b(@)6 b(=@)g(r)s Fu(\))23 b(=)1533 1945 y Fo(d)p 1509 1989 V 1509 2081 a(d\022)1622 2012 y(g)t Fu(\()p Fo(@)6 b(=@)g(r)m(;)17 b(@)6 b(=@)g(s)p Fu(\))23 b(=)2387 1945 y Fo(d)p 2362 1989 V 2362 2081 a(d\022)2475 2012 y(g)t Fu(\()p Fo(@)6 b(=@)g(s;)17 b(@)6 b(=@)g(s)p Fu(\))23 b(=)28 b(0)515 b(\(2\))0 2281 y(on)27 b(\(\006)243 2296 y Fk(1)296 2281 y Fm([)9 b Fu(\006)443 2296 y Fk(2)488 2281 y Fu(\))g Fm(\\)g Fo(B)c Fu(.)42 b(W)-8 b(e)26 b(sa)m(w)h(in)g(a)f(previous)h (paragraph)g(that)g Fo(g)i Fu(mak)m(es)d Fo(@)6 b(=@)g(\022)28 b Fu(and)f Fo(@)6 b(=@)g(s)25 b Fu(orthogonal.)0 2401 y(Recall)f(that)f Fo(@)6 b(=@)g(s)23 b Fu(and)h(the)g(pushforw)m(ard)i (\()p Fo(F)14 b Fu(\))1799 2416 y Fi(\003)1844 2401 y Fo(@)6 b(=@)g(r)26 b Fu(of)e Fo(@)6 b(=@)g(r)25 b Fu(b)m(y)f Fo(F)38 b Fu(are)24 b(orthonormal)e(with)i(resp)s(ect)0 2520 y(to)33 b(the)g(metric)g Fo(g)t Fu(,)f(and)h(observ)m(e)h(that)882 2802 y(\()p Fo(F)14 b Fu(\))1038 2817 y Fi(\003)1119 2735 y Fo(@)p 1095 2779 107 4 v 1095 2871 a(@)6 b(r)1213 2802 y Fu(\()p Fo(r)m(;)17 b(\022)s(;)g(s)p Fu(\))25 b(=)1686 2735 y Fo(@)p 1662 2779 V 1662 2871 a(@)6 b(r)1780 2802 y Fu(\()p Fo(r)m(;)17 b(\022)s(;)g(s)p Fu(\))j(+)j Fo(c)p Fu(\()p Fo(r)m(;)17 b(\022)s(;)g(s)p Fu(\))2591 2735 y Fo(@)p 2569 2779 108 4 v 2569 2871 a(@)6 b(\022)2686 2802 y Fu(\()p Fo(r)m(;)17 b(\022)s(;)g(s)p Fu(\))p Fo(;)0 3071 y Fu(where)30 b Fo(c)p Fu(\()p Fo(r)m(;)17 b(\022)s(;)g(s)p Fu(\))26 b(is)j(a)f(smo)s(oth)g(function)i(that)e(v)-6 b(anishes)30 b(when)g Fo(\022)g Fu(=)e Fo(\022)2651 3086 y Fk(1)2696 3071 y Fo(;)17 b(\022)2788 3086 y Fk(1)2845 3071 y Fu(+)c 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b(in)m(to)g(transv)m(erse)i(in)m (tersections)f(of)g(the)f(relev)-6 b(an)m(t)32 b(stable)g(and)f (unstable)i(manifolds.)42 b(It)0 3847 y(will)32 b(b)s(e)i(used)g(to)f (obtain)g(the)h(horsesho)s(e)g(promised)f(in)g(Theorem)g(0.1.)0 4026 y Fw(Prop)s(osition)42 b(2.4.)58 b Fj(L)-5 b(et)39 b Fo(O)1110 4041 y Fk(1)1193 4026 y Fj(and)g Fo(O)1466 4041 y Fk(2)1550 4026 y Fj(b)-5 b(e)38 b(hyp)-5 b(erb)g(olic)38 b(p)-5 b(erio)g(dic)37 b(orbits)h(for)h(the)f(ge)-5 b(o)g(desic)37 b(\015ow)i(of)f(a)0 4146 y(R)m(iemannian)g(manifold)g Fu(\()p Fo(M)5 b(;)17 b(g)t Fu(\))37 b Fj(that)i(ar)-5 b(e)38 b(c)-5 b(onne)g(cte)g(d)38 b(by)g(a)h(heter)-5 b(o)g(clinic)37 b(orbit)h Fo(O)3199 4161 y Fk(1)p Fn(;)p Fk(2)3340 4146 y Fm(\032)c Fo(W)3559 4109 y Fn(u)3611 4146 y Fu(\()p Fo(O)3726 4161 y Fk(1)3771 4146 y Fu(\))24 b Fm(\\)0 4265 y Fo(W)108 4229 y Fn(s)151 4265 y Fu(\()p Fo(O)266 4280 y Fk(2)310 4265 y Fu(\))p Fj(.)58 b(Then)40 b(we)f(c)-5 b(an)40 b(make)g(an)g(arbitr)-5 b(arily)39 b(smal)5 b(l)41 b(\(in)e(the)h Fo(C)2612 4229 y Fi(1)2736 4265 y Fj(top)-5 b(olo)g(gy\))40 b(p)-5 b(erturb)g(ation)40 b(of)f Fo(g)0 4385 y Fj(in)f(an)f(arbitr)-5 b(arily)38 b(smal)5 b(l)39 b(neighb)-5 b(ourho)g(o)g(d)37 b(of)g(any)i(given)e(p) -5 b(oint)37 b(on)h(the)g(ge)-5 b(o)g(desic)36 b(c)-5 b(orr)g(esp)g(onding)36 b(to)0 4504 y Fo(O)76 4519 y Fk(1)p Fn(;)p Fk(2)218 4504 y Fj(so)d(that)h Fo(O)621 4519 y Fk(1)699 4504 y Fj(and)g Fo(O)967 4519 y Fk(2)1045 4504 y Fj(ar)-5 b(e)34 b(hyp)-5 b(erb)g(olic)33 b(close)-5 b(d)33 b(orbits)g(for)g(the)h(new)g(ge)-5 b(o)g(desic)32 b(\015ow)h(and)h Fo(W)3650 4468 y Fn(u)3702 4504 y Fu(\()p Fo(O)3817 4519 y Fk(1)3861 4504 y Fu(\))0 4624 y Fj(and)h Fo(W)301 4588 y Fn(s)344 4624 y Fu(\()p Fo(O)459 4639 y Fk(2)503 4624 y Fu(\))h Fj(now)g(interse)-5 b(ct)34 b(tr)-5 b(ansversal)5 b(ly)36 b(along)g Fo(O)2114 4639 y Fk(1)p Fn(;)p Fk(2)2222 4624 y Fj(.)0 4802 y(Pr)-5 b(o)g(of.)49 b Fu(The)35 b(metho)s(d)e(for)h(making)f(the)h(in)m 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9 bop 0 100 a Fo(O)76 115 y Fn(i)152 100 y Fu(and)43 b(the)g(strong)g(unstable)h(manifold)d(for)i Fo(O)1919 115 y Fn(i)p Fk(+1)2096 100 y Fu(that)g(con)m(tain)57 b(_)-42 b Fo(\015)5 b Fu(\()p Fo(t)p Fu(\).)72 b(It)42 b(is)h(w)m(ell)f(kno)m(wn)h(that)0 219 y(for)d(all)e Fo(t)p Fu(,)j(except)f(for)f(a)g(discrete)i(subset,)h(there)e(are)f (neigh)m(b)s(ourho)s(o)s(ds)i Fo(N)2939 183 y Fn(ss)3058 219 y Fu(and)f Fo(N)3349 183 y Fn(uu)3487 219 y Fu(of)55 b(_)-43 b Fo(\015)5 b Fu(\()p Fo(t)p Fu(\))39 b(in)0 339 y Fo(W)108 303 y Fn(ss)188 339 y Fu(\()p Fo(t)p Fu(\))31 b(and)h Fo(W)633 303 y Fn(uu)732 339 y Fu(\()p Fo(t)p Fu(\))f(resp)s(ectiv)m(ely)g(that)g(pro)6 b(ject)32 b(to)f(smo)s(oth)f (h)m(yp)s(ersurfaces)k Fo(H)3152 303 y Fn(ss)3232 339 y Fu(\()p Fo(t)p Fu(\))d(and)h Fo(H)3660 303 y Fn(uu)3759 339 y Fu(\()p Fo(t)p Fu(\);)0 458 y(the)k(neigh)m(b)s(ourho)s(o)s(ds)h Fo(N)988 422 y Fn(ss)1103 458 y Fu(and)g Fo(N)1391 422 y Fn(uu)1525 458 y Fu(consist)f(of)g(the)g(unit)g(normals)f(to)g 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Fu(\()p Fo(t)p Fu(\))e(in)m(tersect)h(transv)m(ersally)f(at)63 b(_)-42 b Fo(\015)5 b Fu(\()p Fo(t)p Fu(\);)55 b(if)49 b(this)0 1060 y(prop)s(ert)m(y)33 b(holds)g(for)f(one)h(time)f Fo(t)1263 1075 y Fk(0)1340 1060 y Fu(it)g(will)f(hold)i(for)g(all)e (times)h Fo(t)p Fu(.)44 b(The)33 b(in)m(tersection)g(of)g Fo(W)3469 1023 y Fn(ss)3549 1060 y Fu(\()p Fo(t)3624 1075 y Fk(0)3668 1060 y Fu(\))f(and)0 1179 y Fo(W)108 1143 y Fn(uu)207 1179 y Fu(\()p Fo(t)282 1194 y Fk(0)327 1179 y Fu(\))i(at)49 b(_)-43 b Fo(\015)5 b Fu(\()p Fo(t)655 1194 y Fk(0)699 1179 y Fu(\))35 b(is)f(transv)m(ersal)h(if)g(and)g (only)f(if)g(0)h(is)f(not)h(an)f(eigen)m(v)-6 b(alue)36 b(of)f(the)g(quadratic)f(form)0 1299 y Fo(U)79 1263 y Fn(uu)178 1299 y Fu(\()p Fo(t)253 1314 y Fk(0)297 1299 y Fu(\))22 b Fm(\000)h Fo(U)537 1263 y Fn(ss)617 1299 y Fu(\()p Fo(t)692 1314 y Fk(0)736 1299 y Fu(\).)120 1422 y(No)m(w)45 b Fo(U)438 1386 y Fn(ss)518 1422 y Fu(\()p Fo(t)p Fu(\))h(and)g Fo(U)963 1386 y Fn(uu)1062 1422 y Fu(\()p Fo(t)p Fu(\))g(are)g(solutions)g(of)g(a)g(di\013eren)m(tial)h (equation.)82 b(Supp)s(ose)47 b(that)f Fo(E)3754 1437 y Fn(i)3787 1422 y Fu(\()p Fo(t)p Fu(\))0 1541 y(are)k(co)m(v)-6 b(arian)m(tly)49 b(constan)m(t)i(v)m(ector)e(\014elds)i(along)e Fo(\015)55 b Fu(that)50 b(form)f(an)h(orthonormal)e(basis)j(for)f(the)0 1661 y(orthogonal)39 b(complemen)m(t)g(of)55 b(_)-43 b Fo(\015)5 b Fu(\()p Fo(t)p Fu(\))40 b(for)g(ev)m(ery)g Fo(t)p Fu(.)64 b(Let)40 b Fo(R)q Fu(\()p Fo(t)p Fu(\))f(b)s(e)h(the)h (matrix)d(whose)i Fo(ij)6 b Fu('th)40 b(en)m(try)g(is)0 1781 y Fm(h)p Fo(R)q Fu(\()15 b(_)-43 b Fo(\015)t Fu(\()p Fo(t)p Fu(\))p Fo(;)17 b(E)444 1796 y Fn(i)475 1781 y Fu(\()p Fo(t)p Fu(\)\))e(_)-43 b Fo(\015)5 b Fu(\()p Fo(t)p Fu(\))p Fo(;)17 b(E)918 1796 y Fn(j)958 1781 y Fu(\()p Fo(t)p Fu(\))p Fm(i)p Fu(,)25 b(where)h Fo(R)f Fu(is)g(the)g(curv)-6 b(ature)25 b(tensor.)42 b(Then)26 b(the)f(matrices)f(that)g(express)0 1900 y Fo(U)79 1864 y Fn(ss)159 1900 y Fu(\()p Fo(t)p Fu(\))33 b(and)g Fo(U)578 1864 y Fn(uu)678 1900 y Fu(\()p Fo(t)p Fu(\))f(in)h(terms)g(of)g(this)h (basis)f(satisfy)g(the)h(follo)m(wing)f(matrix)e(Riccati)i(equation:) 1399 2151 y Fo(U)1478 2110 y Fi(0)1506 2151 y Fu(\()p Fo(t)p Fu(\))22 b(+)h Fo(U)1821 2110 y Fk(2)1865 2151 y Fu(\()p Fo(t)p Fu(\))f(+)g Fo(R)q Fu(\()p Fo(t)p Fu(\))27 b(=)h(0)p Fo(:)120 2405 y Fu(What)47 b(is)g(crucial)h(is)f(not)g(the)h (exact)f(form)f(of)i(the)g(equation,)i(but)d(the)h(fact)g(that)f Fo(U)3548 2369 y Fn(ss)3628 2405 y Fu(\()p Fo(t)3703 2420 y Fk(0)3747 2405 y Fu(\))g(is)0 2525 y(determined)29 b(b)m(y)f Fo(R)q Fu(\()p Fo(t)p Fu(\))g(for)g Fo(t)g(<)g(t)1220 2540 y Fk(0)1293 2525 y Fu(and)h Fo(U)1561 2488 y Fn(uu)1660 2525 y Fu(\()p Fo(t)1735 2540 y Fk(0)1779 2525 y Fu(\))g(is)f (determined)h(b)m(y)f Fo(R)q Fu(\()p Fo(t)p Fu(\))f(for)i Fo(t)f(>)g(t)3162 2540 y Fk(0)3207 2525 y Fu(.)42 b(The)29 b(orbit)f Fo(O)3792 2540 y Fk(1)p Fn(;)p Fk(2)0 2644 y Fu(is)g(nonrecurren)m(t,)j(b)s(ecause)f(it)e(is)g(forw)m(ards)h(and)g 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b(in)0 3601 y(a)h(arbitr)-5 b(arily)35 b(smal)5 b(l)37 b(neighb)-5 b(orho)g(o)g(d)34 b(of)h(the)g(p)-5 b(oint)35 b Fo(\015)6 b Fu(\()p Fo(t)2083 3616 y Fk(0)2127 3601 y Fu(\))p Fj(.)1626 b Fg(\004)120 3791 y Fu(The)22 b(\014nal)h(prop)s(osition)f(allo)m(ws)f (us)i(to)f(glue)g(together)g(geo)s(desics)h(that)f(are)h(forw)m(ard)f (and)h(bac)m(kw)m(ards)0 3910 y(asymptotic)32 b(to)g(a)h(h)m(yp)s(erb)s (olic)h(closed)g(geo)s(desic.)0 4100 y Fw(Prop)s(osition)k(2.5.)58 b Fj(L)-5 b(et)36 b Fo(\015)41 b Fj(b)-5 b(e)35 b(a)g(close)-5 b(d)36 b(ge)-5 b(o)g(desic)34 b(in)h(a)g(R)m(iemannian)g(manifold)h Fu(\()p Fo(M)5 b(;)17 b(g)t Fu(\))33 b Fj(for)j(which)0 4220 y(the)46 b(c)-5 b(orr)g(esp)g(onding)45 b(orbit)g Fo(O)k Fj(of)d(the)g(ge)-5 b(o)g(desic)45 b(\015ow)h(is)g(hyp)-5 b(erb)g(olic.)76 b(L)-5 b(et)47 b Fo(v)k Fm(2)c Fo(W)3263 4184 y Fn(s)3306 4220 y Fu(\()p Fo(O)s Fu(\))e Fj(and)i Fo(w)j Fm(2)0 4339 y Fo(W)108 4303 y Fn(u)160 4339 y Fu(\()p Fo(O)s Fu(\))p Fj(.)44 b(Then)34 b(we)g(c)-5 b(an)34 b(make)g(an)g(arbitr)-5 b(arily)34 b(smal)5 b(l)35 b(\(in)e(the)h Fo(C)2514 4303 y Fi(1)2633 4339 y Fj(top)-5 b(olo)g(gy\))34 b(p)-5 b(erturb)g(ation)34 b(of)f Fo(g)k Fj(in)0 4459 y(an)32 b(arbitr)-5 b(arily)31 b(smal)5 b(l)34 b(neighb)-5 b(ourho)g(o)g(d)30 b(of)i(any)g(given)f(p)-5 b(oint)31 b(on)h Fo(\015)37 b Fj(so)32 b(that)g Fo(v)k Fj(and)c Fo(w)j Fj(lie)c(on)h(the)g(same)0 4579 y(orbit)j(of)g(the)h (new)f(ge)-5 b(o)g(desic)34 b(\015ow.)0 4769 y(Pr)-5 b(o)g(of.)49 b Fu(Let)32 b Fo(N)42 b Fu(b)s(e)31 b(a)g(neigh)m(b)s (ourho)s(o)s(d)h(of)g(the)f(giv)m(en)g(p)s(oin)m(t)h(on)f Fo(\015)6 b Fu(.)42 b(W)-8 b(e)32 b(can)f(c)m(ho)s(ose)h Fo(t)3267 4784 y Fk(1)3340 4769 y Fo(>)c Fu(0)f Fo(>)i(t)3664 4784 y Fk(2)3739 4769 y Fu(and)0 4888 y Fo(\016)36 b(>)d Fu(0)i(suc)m(h)j(that,)d(if)h Fo(N)923 4903 y Fk(1)1004 4888 y Fu(and)g Fo(N)1280 4903 y Fk(2)1361 4888 y Fu(are)g(the)g (closed)g Fo(\016)t Fu(-discs)h(around)g Fo(p)2714 4903 y Fk(1)2791 4888 y Fu(=)32 b Fo(\015)2952 4903 y Fn(v)2999 4888 y Fu(\()p Fo(t)3074 4903 y Fk(1)3119 4888 y Fu(\))j(and)h Fo(p)3439 4903 y Fk(2)3516 4888 y Fu(=)d Fo(\015)3678 4903 y Fn(w)3742 4888 y Fu(\()p Fo(t)3817 4903 y Fk(2)3861 4888 y Fu(\))0 5008 y(resp)s(ectiv)m(ely)-8 b(,)33 b(then)152 5161 y(\(1\))49 b Fo(N)409 5176 y Fk(1)476 5161 y Fm(\\)22 b Fo(N)644 5176 y Fk(2)717 5161 y Fu(=)28 b Fm(;)33 b Fu(and)h Fo(N)1179 5176 y Fk(1)1245 5161 y Fm([)23 b Fo(N)1414 5176 y Fk(2)1486 5161 y Fm(\032)29 b Fo(N)11 b Fu(;)152 5280 y(\(2\))49 b Fo(\015)381 5295 y Fn(v)428 5280 y Fu(\()p Fo(t)p Fu(\))38 b Fo(=)-61 b Fm(2)28 b Fo(N)743 5295 y Fk(1)810 5280 y Fm([)22 b Fo(N)978 5295 y Fk(2)1056 5280 y Fu(for)34 b Fo(t)27 b Fm(\025)i Fu(0)k(except)g(for) h Fo(t)1962 5295 y Fk(1)2029 5280 y Fm(\000)22 b Fo(\016)32 b Fm(\024)c Fo(t)g Fm(\024)g Fo(t)2514 5295 y Fk(1)2581 5280 y Fu(+)22 b Fo(\016)38 b Fu(when)c Fo(\015)3074 5295 y Fn(v)3121 5280 y Fu(\()p Fo(t)p Fu(\))27 b Fm(2)h Fo(N)3436 5295 y Fk(1)3481 5280 y Fu(;)152 5400 y(\(3\))49 b Fo(\015)381 5415 y Fn(w)445 5400 y Fu(\()p Fo(t)p Fu(\))38 b Fo(=)-61 b Fm(2)28 b Fo(N)760 5415 y Fk(1)827 5400 y Fm([)23 b Fo(N)996 5415 y Fk(2)1073 5400 y Fu(for)34 b Fo(t)28 b Fm(\024)g Fu(0)33 b(except)g(for)h Fo(t)1979 5415 y Fk(2)2046 5400 y Fm(\000)22 b Fo(\016)32 b Fm(\024)c Fo(t)g Fm(\024)g Fo(t)2531 5415 y Fk(2)2598 5400 y Fu(+)23 b Fo(\016)37 b Fu(when)d Fo(\015)3091 5415 y Fn(w)3155 5400 y Fu(\()p Fo(t)p Fu(\))27 b Fm(2)h Fo(N)3470 5415 y Fk(2)3515 5400 y Fu(.)1908 5639 y Fs(10)p eop %%Page: 11 11 11 10 bop 0 100 a Fu(Since)38 b(the)f(closed)h(orbit)f Fo(O)j Fu(is)d(h)m(yp)s(erb)s(olic,)h(w)m(e)f(can)h(\014nd)g(v)m (ectors)f Fo(v)2659 63 y Fi(0)2724 100 y Fu(and)g Fo(w)2995 63 y Fi(0)3060 100 y Fu(as)h(close)f(as)g(w)m(e)h(wish)0 219 y(to)46 b(_)-43 b Fo(\015)172 234 y Fn(v)219 219 y Fu(\()p Fo(t)294 234 y Fk(1)339 219 y Fu(\))31 b(and)47 b(_)-43 b Fo(\015)653 234 y Fn(w)717 219 y Fu(\()p Fo(t)792 234 y Fk(2)837 219 y Fu(\))31 b(resp)s(ectiv)m(ely)h(suc)m(h)h(that)e Fo(w)1960 183 y Fi(0)2020 219 y Fu(lies)g(on)h(the)g(forw)m(ard)h (orbit)e(of)h Fo(v)3268 183 y Fi(0)3327 219 y Fu(and)g(the)g(orbit)0 339 y(segmen)m(t)j Fo(O)466 303 y Fi(0)529 339 y Fu(b)s(et)m(w)m(een)h Fo(v)967 303 y Fi(0)1030 339 y Fu(and)g Fo(w)1300 303 y Fi(0)1363 339 y Fu(lies)f(as)g(close)g(as)h(wish)f(to)g(the)g(union)h (of)f(the)g(forw)m(ard)h(orbit)f(of)g Fo(v)0 458 y Fu(and)g(the)g(bac)m (kw)m(ard)g(orbit)f(of)g Fo(w)s Fu(.)48 b(In)35 b(particular,)f(w)m(e)h (can)g(ensure)h(that)e Fo(\015)2852 473 y Fn(v)2895 453 y Fe(0)2925 458 y Fu(\()p Fo(\016)t Fu(\))g(and)h Fo(\015)3332 473 y Fn(w)3392 453 y Fe(0)3423 458 y Fu(\()p Fm(\000)p Fo(\016)t Fu(\))g(lie)f(on)0 578 y(the)i(b)s(oundaries)h(of)f Fo(N)880 593 y Fk(1)960 578 y Fu(and)g Fo(N)1236 593 y Fk(2)1317 578 y Fu(resp)s(ectiv)m(ely)g(and)g(the)g(segmen)m(t)f(of)h Fo(\015)2788 593 y Fn(v)2831 573 y Fe(0)2898 578 y Fu(that)f(lies)h(b)s (et)m(w)m(een)h(these)0 697 y(p)s(oin)m(ts)c(do)s(es)h(not)f(en)m(ter)h Fo(N)1033 712 y Fk(1)1100 697 y Fm([)23 b Fo(N)1269 712 y Fk(2)1314 697 y Fu(.)120 819 y(W)-8 b(e)39 b(no)m(w)g(sho)m(w)g(that) g(it)f(is)h(p)s(ossible)g(to)f(p)s(erturb)i(the)f(metric)f(inside)h Fo(N)2932 834 y Fk(1)3016 819 y Fu(so)g(that)f(the)h(geo)s(desic)0 939 y(whic)m(h)44 b(en)m(ters)h Fo(N)677 954 y Fk(1)765 939 y Fu(tangen)m(t)e(to)58 b(_)-43 b Fo(\015)1321 954 y Fn(v)1368 939 y Fu(\()p Fo(t)1443 954 y Fk(1)1517 939 y Fm(\000)29 b Fo(\016)t Fu(\))43 b(exits)g Fo(N)2079 954 y Fk(1)2167 939 y Fu(tangen)m(t)h(to)57 b(_)-42 b Fo(\015)2724 954 y Fn(v)2767 934 y Fe(0)2797 939 y Fu(\()p Fo(\016)t Fu(\).)74 b(In)44 b(order)g(to)f(do)g(this,)0 1058 y(c)m(ho)s(ose)c(a)g(sphere)h Fo(S)k Fu(that)38 b(is)g(close)h(to)f(the)h(geo)s(desic)g(sphere)h(of)f(radius)g Fo(\016)t(=)p Fu(2)f(around)h Fo(p)3420 1073 y Fk(1)3501 1058 y Fu(=)e Fo(\015)3667 1073 y Fn(v)3714 1058 y Fu(\()p Fo(t)3789 1073 y Fk(1)3834 1058 y Fu(\),)0 1178 y(passes)e(through)g Fo(\015)731 1193 y Fn(v)778 1178 y Fu(\()p Fo(t)853 1193 y Fk(1)920 1178 y Fm(\000)23 b Fo(\016)t(=)p Fu(2\))33 b(and)i Fo(\015)1487 1193 y Fn(v)1530 1173 y Fe(0)1561 1178 y Fu(\()p Fo(\016)t(=)p Fu(2\),)e(and)h(is)g(orthogonal)g(to)48 b(_)-43 b Fo(\015)2817 1193 y Fn(v)2864 1178 y Fu(\()p Fo(t)2939 1193 y Fk(1)3006 1178 y Fm(\000)23 b Fo(\016)t(=)p Fu(2\))34 b(and)49 b(_)-43 b Fo(\015)3573 1193 y Fn(v)3616 1173 y Fe(0)3647 1178 y Fu(\()p Fo(\016)t(=)p Fu(2\).)0 1297 y(Let)46 b Fo(D)273 1312 y Fn(r)363 1297 y Fu(b)s(e)g(the)g (closed)h Fo(r)s Fu(-disc)f(and)g Fo(S)1562 1312 y Fn(r)1652 1297 y Fu(the)g(sphere)i(of)d(radius)i Fo(r)h Fu(around)e(the)h(origin) e(in)g Fo(T)3707 1312 y Fn(p)3748 1322 y Fh(1)3792 1297 y Fo(M)0 1417 y Fu(with)39 b(the)h(geometry)d(giv)m(en)i(b)m(y)h(the)f (inner)h(pro)s(duct)g Fo(g)2128 1432 y Fn(p)2169 1442 y Fh(1)2213 1417 y Fu(\()p Fm(\001)p Fo(;)17 b Fm(\001)p Fu(\))37 b(de\014ned)k(on)e Fo(T)2980 1432 y Fn(p)3021 1442 y Fh(1)3066 1417 y Fo(M)49 b Fu(b)m(y)40 b Fo(g)t Fu(.)61 b(Cho)s(ose)39 b(a)0 1536 y(di\013eomorphism)31 b Fo( )h Fu(:)27 b Fo(D)934 1554 y Fn(\016)s(=)p Fk(2)1086 1536 y Fm(!)h Fo(N)1294 1551 y Fk(1)1371 1536 y Fu(that)k(is)h(close)g (to)f(the)i(exp)s(onen)m(tial)e(map)g(and)h(has)g(the)h(follo)m(wing)0 1656 y(prop)s(erties:)152 1807 y(\(1\))49 b(The)34 b(initial)d(v)-6 b(alue)34 b Fo( )t Fu(\(0\))26 b(=)i Fo(p)1457 1822 y Fk(1)1502 1807 y Fu(.)152 1927 y(\(2\))49 b(The)34 b(map)e Fo( )37 b Fu(satis\014es)d Fo( )t Fu(\()p Fo(@)6 b(D)1473 1945 y Fn(\016)s(=)p Fk(2)1596 1927 y Fu(\))27 b(=)i Fo(S)39 b Fu(and)33 b Fo( )t Fu(\(In)m(t)16 b Fo(D)2394 1945 y Fn(\016)s(=)p Fk(2)2519 1927 y Fu(\))33 b(lies)g(inside)h Fo(S)6 b Fu(.)152 2047 y(\(3\))49 b(The)29 b(map)f Fo( )k Fu(maps)c(one)h(diagonal)e(of)i Fo(D)1861 2065 y Fn(\016)s(=)p Fk(2)2015 2047 y Fu(to)f(a)g(curv)m(e)h(joining)g Fo(\015)2848 2062 y Fn(v)2895 2047 y Fu(\()p Fo(t)2970 2062 y Fk(1)3027 2047 y Fm(\000)13 b Fo(\016)t(=)p Fu(2\))28 b(and)h Fo(\015)3573 2062 y Fn(v)3616 2042 y Fe(0)3647 2047 y Fu(\()p Fo(\016)t(=)p Fu(2\).)152 2167 y(\(4\))49 b(The)35 b(map)f Fo(D)s( )39 b Fu(maps)34 b(the)h(in)m(w)m(ard)g(unit)g(normal)f(v)m(ector)g (\014eld)i(on)f Fo(@)6 b(D)37 b Fu(to)d(the)h(in)m(w)m(ard)h(unit)329 2287 y(normal)c(v)m(ector)h(\014eld)h(on)f Fo(S)6 b Fu(.)152 2406 y(\(5\))49 b(If)39 b Fo(z)j Fm(2)c Fo(@)6 b(D)s Fu(,)40 b(then)g(the)g(curv)m(e)f Fo(t)f Fm(7!)f Fo( )t Fu(\(\(1)25 b Fm(\000)h Fo(t)p Fu(\))p Fo(z)t Fu(\),)41 b(0)c Fm(\024)h Fo(t)f Fm(\024)h Fu(1)p Fo(=)p Fu(2)h(is)g(the)g(geo)s (desic)h(segmen)m(t)329 2526 y(in)c Fo(N)528 2541 y Fk(1)572 2526 y Fu(,)g(parametrized)g(with)f(sp)s(eed)j Fo(\016)t(=)p Fu(2,)d(that)h(starts)g(orthogonally)e(from)h Fo(S)42 b Fu(at)35 b Fo( )t Fu(\()p Fo(z)t Fu(\))g(and)329 2645 y(go)s(es)e(distance)h Fo(\016)t(=)p Fu(4)f(in)m(to)g(the)g(in)m (terior)g(of)h Fo(S)6 b Fu(.)120 2797 y(Let)29 b Fo(V)52 b Fu(b)s(e)29 b(the)h(v)m(ector)f(\014eld)h(on)g Fo(D)1427 2815 y Fn(\016)s(=)p Fk(2)1566 2797 y Fm(n)14 b(f)p Fu(0)p Fm(g)29 b Fu(that)g(p)s(oin)m(ts)g(radially)f(out)m(w)m(ard)i(and)g (has)f(unit)h(length)0 2917 y(with)44 b(resp)s(ect)i(to)f Fo(g)773 2932 y Fn(p)814 2942 y Fh(1)857 2917 y Fu(.)79 b(Prop)s(erties)45 b(\(4\))f(and)h(\(5\))f(of)h Fo( )j Fu(ensure)f(that)d(if)h Fo(z)52 b Fm(2)47 b Fo(D)3190 2935 y Fn(\016)s(=)p Fk(2)3345 2917 y Fm(n)29 b Fo(D)3506 2935 y Fn(\016)s(=)p Fk(4)3631 2917 y Fu(,)47 b(then)0 3037 y Fo(D)s( )t Fu(\()p Fo(V)22 b Fu(\()p Fo(z)t Fu(\)\))33 b(is)g(a)g(unit)g(v)m(ector)h(orthogonal)e(to)h Fo( )t Fu(\()p Fo(S)1955 3055 y Fi(k)p Fn(z)s Fi(k)2082 3037 y Fu(\).)120 3159 y(W)-8 b(e)43 b(can)h(de\014ne)h(a)e(new)h (Riemannian)f(metric)g(on)g Fo( )t Fu(\()p Fo(D)2324 3177 y Fn(\016)s(=)p Fk(2)2477 3159 y Fm(n)29 b(f)p Fu(0)p Fm(g)p Fu(\))43 b(b)m(y)g(lea)m(ving)g(the)h(lengths)g(of)0 3280 y(v)m(ectors)d(tangen)m(t)g(to)f(the)h(spheres)h Fo( )t Fu(\()p Fo(S)1550 3295 y Fn(r)1594 3280 y Fu(\))e(unc)m(hanged)j (and)e(decreeing)h(that)e Fo(D)s( )t Fu(\()p Fo(V)21 b Fu(\()p Fo(z)t Fu(\)\))41 b(is)g(a)f(unit)0 3399 y(v)m(ector)28 b(\014eld)h(that)f(is)g(orthogonal)f(to)h(these)h(spheres.)44 b(This)28 b(metric)f(agrees)i(with)f(the)g(original)f(metric)0 3519 y(outside)42 b(the)f(image)f(of)i Fo( )t Fu(\()p Fo(D)1139 3537 y Fn(\016)s(=)p Fk(4)1263 3519 y Fu(\))f(and)g(extends)i (smo)s(othly)c(to)i(a)g(metric)f(on)h Fo( )t Fu(\()p Fo(D)3231 3537 y Fn(\016)s(=)p Fk(2)3355 3519 y Fu(\).)68 b(It)41 b(follo)m(ws)0 3639 y(from)35 b(Gauss')i(Lemma)d(\(see)j(e.g.) 52 b(Lemma)34 b(3.3.5)h(in)h([DoC]\))f(that)g Fo( )40 b Fu(maps)35 b(the)h(diagonals)g(of)g Fo(D)j Fu(to)0 3758 y(unit)33 b(sp)s(eed)i(geo)s(desics)f(in)f Fo(N)1111 3773 y Fk(1)1189 3758 y Fu(with)g(its)g(new)h(metric.)120 3880 y(Finally)k(w)m(e)i(mak)m(e)f(an)h(analogous)f(c)m(hange)i(of)f (metric)e(inside)i Fo(N)2637 3895 y Fk(2)2722 3880 y Fu(so)g(that)f(the)h(geo)s(desic)g(whic)m(h)0 4000 y(en)m(ters)34 b Fo(N)371 4015 y Fk(2)449 4000 y Fu(tangen)m(t)g(to)47 b(_)-42 b Fo(\015)986 4015 y Fn(w)1046 3995 y Fe(0)1077 4000 y Fu(\()p Fm(\000)p Fo(\016)t Fu(\))33 b(exits)g Fo(N)1629 4015 y Fk(2)1707 4000 y Fu(tangen)m(t)g(to)48 b(_)-43 b Fo(\015)2243 4015 y Fn(w)2307 4000 y Fu(\()p Fo(t)2382 4015 y Fk(2)2449 4000 y Fu(+)22 b Fo(\016)t Fu(\).)1160 b Fg(\004)713 4249 y Fw(3.)55 b(Construction)35 b(of)i(an)f(arbitrarily)g(dense)f(horsesho)s(e)120 4431 y Fu(W)-8 b(e)49 b(b)s(egin)h(b)m(y)f(reminding)g(the)g(reader)h(of)g (some)f(facts)g(ab)s(out)h(the)f(geometry)f(of)h(the)h(round)0 4550 y(sphere)35 b Fo(S)378 4514 y Fk(3)422 4550 y Fu(,)e(whic)m(h)h(w) m(e)g(collect)f(in)m(to)g(Lemma)e(3.1.)0 4736 y Fw(Lemma)50 b(3.1.)58 b Fj(Two)45 b(distinct)f(close)-5 b(d)45 b(ge)-5 b(o)g(desics)43 b(on)i Fo(S)2204 4700 y Fk(3)2293 4736 y Fj(interse)-5 b(ct)44 b(if)g(and)h(only)h(if)e(they)i(lie)e(in)h(a)0 4856 y(c)-5 b(ommon)39 b(gr)-5 b(e)g(at)39 b(spher)-5 b(e.)54 b(If)38 b(they)h(interse)-5 b(ct,)38 b(they)i(lie)e(on)h(a)g (unique)g(2-spher)-5 b(e,)37 b(interse)-5 b(ct)38 b(at)h(a)g(p)-5 b(air)0 4975 y(of)35 b(antip)-5 b(o)g(dal)35 b(p)-5 b(oints)35 b(and)h(have)f(a)g(c)-5 b(ommon)36 b(normal)h(dir)-5 b(e)g(ction.)0 5161 y(Pr)g(o)g(of.)49 b Fu(The)41 b(pro)s(of)e(is)h(an) g(immediate)d(consequence)42 b(of)e(the)g(facts)h(that)e(a)g(closed)i (geo)s(desic)f(is)g(the)0 5280 y(in)m(tersection)32 b(of)g(a)f(plane)h (through)g(the)f(origin)g(with)g Fo(S)2107 5244 y Fk(3)2183 5280 y Fu(and)h(a)f(great)g(sphere)i(is)f(the)g(in)m(tersection)g(of)0 5400 y(a)h(h)m(yp)s(erplane)h(through)g(the)f(origin)g(with)g Fo(S)1720 5364 y Fk(3)1764 5400 y Fu(.)2031 b Fg(\004)1908 5639 y Fs(11)p eop %%Page: 12 12 12 11 bop 120 100 a Fu(It)41 b(is)g(eviden)m(t)h(from)e(the)i(pro)s(of) f(of)h(this)f(lemma)e(that)i(it)g(is)g(exceptional)g(for)h(t)m(w)m(o)f (geo)s(desics)h(to)0 219 y(in)m(tersect.)120 401 y(By)26 b(a)h Fw(sequence)j(of)i(orthogonal)g(spheres)p Fu(,)27 b(w)m(e)h(will)e(mean)h(a)g(sequence)i(of)f(great)f(2-)h(spheres)h(in)0 520 y Fo(S)67 484 y Fk(3)144 520 y Fu(with)j(the)h(prop)s(ert)m(y)g 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b(Let)37 b Fo(\033)1389 1253 y Fn(i)1458 1238 y Fu(b)s(e)f(the)g(great)g(circle) g(in)g(whic)m(h)h(\006)2770 1253 y Fn(i)p Fi(\000)p Fk(1)2941 1238 y Fu(and)g(\006)3210 1253 y Fn(i)3279 1238 y Fu(in)m(tersect.)53 b(The)0 1358 y(circles)37 b(formed)g(in)g(this)g(w)m(a)m(y)g(will)f(b)s (e)h(called)g(the)g Fw(great)42 b(circles)f(asso)s(ciated)h(with)35 b Fm(f)p Fu(\006)3540 1373 y Fn(i)3573 1358 y Fm(g)3623 1373 y Fn(i)p Fi(2)p Fn(I)3751 1358 y Fu(.)55 b(If)0 1478 y(the)34 b(sequence)g(is)g(nondegenerate,)g(then)152 1628 y(\(1\))49 b Fo(\033)386 1643 y Fn(i)452 1628 y Fu(and)34 b Fo(\033)703 1643 y Fn(j)778 1628 y Fu(are)f(distinct)h (unless)g(unless)g Fo(i)28 b Fu(=)h Fo(j)6 b Fu(.)152 1747 y(\(2\))49 b(If)33 b Fo(i)28 b Fm(6)p Fu(=)h Fo(j)6 b Fu(,)32 b(then)i Fo(\033)987 1762 y Fn(i)1053 1747 y Fu(in)m(tersects)h Fo(\033)1552 1762 y Fn(j)1627 1747 y Fu(if)e(and)h(only)f(if)g Fo(i)g Fu(and)h Fo(j)39 b Fu(are)33 b(consecutiv)m(e)i(in)m(tegers.)120 1898 y(A)27 b(sequence)j(of)e(orthogonal)f(spheres)j(in)e(whic)m(h)h(all)e(terms)h (are)g(distinct)g(can)g(b)s(e)h(made)e(nondegen-)0 2017 y(erate)f(\(without)g(destro)m(ying)g(the)h(orthogonalit)m(y)e(prop)s (ert)m(y\))h(b)m(y)g(an)h(arbitrarily)d(small)h(p)s(erturbation.)0 2137 y(In)33 b(order)h(to)e(see)i(this,)f(observ)m(e)h(that)e(after)i (making)d(an)m(y)i(small)e(p)s(erturbation)j(to)e(one)i(term)e(\006) 3693 2152 y Fn(i)3726 2137 y Fu(,)h(w)m(e)0 2256 y(can)i(main)m(tain)f (the)h(orthogonalit)m(y)e(of)i(consecutiv)m(e)h(terms)e(b)m(y)g(p)s (erturbing)i(only)e(the)h(t)m(w)m(o)g(adjacen)m(t)0 2376 y(terms)28 b(\006)345 2391 y Fn(i)p Fi(\000)p Fk(1)509 2376 y Fu(and)h(\006)770 2391 y Fn(i)p Fk(+1)904 2376 y Fu(.)43 b(Since)30 b(all)e(terms)g(in)h(the)g(sequence)i(are)e (distinct,)g(\006)2944 2391 y Fn(i)2991 2376 y Fm(\\)13 b Fu(\006)3142 2391 y Fn(i)p Fi(\000)p Fk(2)3306 2376 y Fu(and)30 b(\006)3568 2391 y Fn(i)3614 2376 y Fm(\\)13 b Fu(\006)3765 2391 y Fn(i)p Fk(+2)0 2495 y Fu(are)24 b(b)s(oth)g(great)g(circles.)42 b(When)25 b(w)m(e)g(mak)m(e)e(a)h (small)f(enough)i(p)s(erturbation)f(to)g(\006)3055 2510 y Fn(i)3088 2495 y Fu(,)i(these)f(great)e(circles)0 2615 y(mo)m(v)m(e)39 b(to)f(nearb)m(y)i(great)f(circles,)j(and)d(w)m(e)h (can)g(c)m(ho)s(ose)g(the)g(new)g(\006)2645 2630 y Fn(i)p Fi(\000)p Fk(1)2819 2615 y Fu(and)g(\006)3091 2630 y Fn(i)p Fk(+1)3265 2615 y Fu(to)f(b)s(e)g(orthogo-)0 2734 y(nal)33 b(to)g(these)h(new)f(great)g(circles)h(\(whic)m(h)f(ensures)i (that)e(the)h(sequence)g(still)f(has)g(the)h(orthogonalit)m(y)0 2854 y(prop)s(ert)m(y\))h(and)g(close)g(to)g(the)g(old)f(\006)1419 2869 y Fn(i)p Fi(\000)p Fk(1)1589 2854 y Fu(and)i(\006)1857 2869 y Fn(i)p Fk(+1)1991 2854 y Fu(.)48 b(W)-8 b(e)36 b(can)f(mak)m(e)f(the)h(sequence)h(nondegenerate)0 2973 y(b)m(y)d(iterating)g(this)g(construction.)120 3094 y(In)c(the)h(pro)s (ofs)g(of)g(Theorems)f(0.1)g(and)h(0.2)e(w)m(e)i(need)h(to)e(able)g(to) g(start)g(with)g(a)g(giv)m(en)h(collection)f(of)0 3213 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b(terms)g(in)i(this)f (sequence)i(will)d(b)s(e)h(distinct)g(if)h(the)f(linking)f(spheres)j (are)e(c)m(hosen)i(suitably)-8 b(.)0 4528 y(If)37 b(necessary)-8 b(,)39 b(it)e(can)g(then)h(b)s(e)f(made)g(nondegenerate)i(b)m(y)e(an)g (arbitrarily)e(small)h(p)s(erturbation,)i(as)0 4648 y(describ)s(ed)d (ab)s(o)m(v)m(e.)0 4830 y Fw(Pro)s(of)49 b(of)e(Theorem)g(0.1.)57 b Fu(W)-8 b(e)42 b(b)s(egin)f(b)m(y)h(c)m(ho)s(osing)f(a)g(\014nite)h (n)m(um)m(b)s(er)g(of)f(\(necessarily)h(closed\))0 4949 y(orbits)35 b(of)g(the)g(geo)s(desic)h(\015o)m(w)f(of)g(the)h(usual)f (round)h(metric)e(on)h Fo(S)2535 4913 y Fk(3)2614 4949 y Fu(whose)h(union)g(is)e Fo(")p Fu(-dense)k(in)d(the)0 5069 y(unit)30 b(tangen)m(t)g(bundle)i(\(with)d(resp)s(ect)i(to)e(the)i (round)g(metric\).)41 b(Let)30 b Fo(c)2666 5084 y Fk(1)2711 5069 y Fo(;)17 b(:)g(:)g(:)d(;)j(c)2976 5084 y Fn(m)3081 5069 y Fu(b)s(e)31 b(the)f(great)f(circles)0 5188 y(to)24 b(whic)m(h)i(these)f(orbits)g(are)g(tangen)m(t.)41 b(W)-8 b(e)25 b(ma)m(y)e(assume)i(that)f(the)i(orbits)e(are)h(all)f(tangen)m (t)h(to)f(distinct)p 0 5304 598 4 v 120 5372 a Fd(1)161 5400 y Fs(If)29 b(the)f(c)n(hain)g(has)h Fr(k)d Fp(\024)d Fs(3)29 b(terms,)e(w)n(e)h(require)f(that)i(they)e(in)n(tersect)h(in)h (great)f(sphere)g(of)h(dimension)g(3)19 b Fp(\000)g Fr(k)r Fs(.)1908 5639 y(12)p eop %%Page: 13 13 13 12 bop 0 100 a Fu(circles.)56 b(Cho)s(ose)38 b(a)f(cross)h Fo(X)1120 115 y Fn(k)1205 100 y Fu(cen)m(tered)h(at)e(eac)m(h)h Fo(c)2002 115 y Fn(k)2089 100 y Fu(so)f(that)g(the)g(4)p Fo(m)h Fu(spheres)h(in)e(the)g(crosses)i(are)0 219 y(all)h(distinct.)67 b(By)40 b(linking)g(consecutiv)m(e)i(terms)f(of)g(the)g(cyclic)g (sequence)i Fo(X)2975 234 y Fk(1)3019 219 y Fo(;)17 b(:)g(:)g(:)d(;)j (X)3324 234 y Fn(m)3398 219 y Fu(,)43 b(w)m(e)e(obtain)0 339 y(a)g(cyclic)f(sequence)i(of)f(orthogonal)f(spheres)j(with)d(3)p Fo(m)h Fu(terms)f(\006)2546 354 y Fk(1)2591 339 y Fo(;)17 b(:)g(:)g(:)d(;)j Fu(\006)2885 354 y Fk(3)p Fn(m)3000 339 y Fu(.)66 b(It)41 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b(orbits)g(of)g(the)h(geo)s(desic)f(\015o)m(w)h(formed)f(b)m(y)g(unit)g (v)m(ectors)g(tangen)m(t)h(to)e Fo(\033)3036 832 y Fn(i)3069 817 y Fu(.)74 b(W)-8 b(e)43 b(c)m(ho)s(ose)h(these)0 936 y(directions)38 b(of)f Fo(O)652 951 y Fk(1)697 936 y Fo(;)17 b(:)g(:)g(:)d(;)j(O)995 951 y Fk(3)p Fn(m)1146 936 y Fu(so)38 b(that)e(these)i(orbits)f(include)h(the)g Fo(m)f Fu(orbits)g(with)g(whic)m(h)h(w)m(e)g(b)s(egan)0 1056 y(the)c(construction.)120 1176 y(F)-8 b(or)33 b(eac)m(h)i Fo(i)29 b Fm(2)h Fc(Z)p Fo(=)p Fu(3)p Fo(m)p Fu(,)h(let)j Fo(p)1193 1191 y Fn(i)1260 1176 y Fu(b)s(e)h(one)f(of)h(the)f(t)m(w)m (o)g(an)m(tip)s(o)s(dal)f(p)s(oin)m(ts)i(in)f(whic)m(h)g(the)h(great)e (circles)0 1295 y Fo(\033)57 1310 y Fn(i)127 1295 y Fu(and)38 b Fo(\033)382 1310 y Fn(i)p Fk(+1)553 1295 y Fu(in)m(tersect.)56 b(Then)39 b Fo(p)1320 1310 y Fn(i)1387 1295 y Fm(2)c Fu(\006)1560 1310 y Fn(i)p Fi(\000)p Fk(1)1719 1295 y Fm(\\)26 b Fu(\006)1883 1310 y Fn(i)1940 1295 y Fm(\\)g Fu(\006)2104 1310 y Fn(i)p Fk(+1)2275 1295 y Fu(and)37 b(it)g(follo)m(ws)g(from)f(\(2\))h(ab)s(o)m(v)m(e)g(that)g Fo(p)3867 1310 y Fn(i)0 1415 y Fu(is)32 b(the)h(cen)m(ter)g(of)g(a)f (ball)g Fo(B)1029 1430 y Fn(i)1094 1415 y Fu(whic)m(h)h(do)s(es)g(not)f (in)m(tersect)h(an)m(y)f(other)h(\006)2698 1430 y Fn(j)2740 1415 y Fu(.)43 b(In)33 b(particular)f Fo(\033)3451 1430 y Fn(i)3516 1415 y Fu(and)h Fo(\033)3766 1430 y Fn(i)p Fk(+1)0 1534 y Fu(are)g(the)h(only)f Fo(\033)614 1549 y Fn(j)655 1534 y Fu('s)h(that)e(en)m(ter)j Fo(B)1300 1549 y Fn(i)1332 1534 y Fu(.)120 1654 y(W)-8 b(e)28 b(no)m(w)h(describ) s(e)g(the)g(sequence)g(of)g(p)s(erturbations)f(that)g(pro)s(duces)i (the)e(desired)i(metric.)41 b(Inside)0 1773 y(eac)m(h)29 b Fo(B)295 1788 y Fn(i)356 1773 y Fu(w)m(e)g(apply)f(Prop)s(osition)f (2.2)h(to)f(mak)m(e)g Fo(O)1901 1788 y Fn(i)1963 1773 y Fu(and)h Fo(O)2227 1788 y Fn(i)p Fk(+1)2390 1773 y Fu(h)m(yp)s(erb)s(olic)g(as)g(orbits)g(of)h(the)f(geo)s(desic)0 1893 y(\015o)m(w)36 b(for)f(\006)436 1908 y Fn(i)505 1893 y Fu(and)h(create)g(a)f(hetero)s(clinic)g(orbit)g(connecting)i (them)e(in)g(the)h(unit)f(tangen)m(t)h(bundle)h(of)0 2012 y(\006)72 2027 y Fn(i)105 2012 y Fu(.)43 b(Note)30 b(that)f(the)i(c)m(hange)g(of)f(metric)f(in)h(Prop)s(osition)f(2.2)h (lea)m(v)m(es)g(\006)2696 2027 y Fn(i)2759 2012 y Fu(totally)e(geo)s (desic,)j(so)f(the)h(set)0 2132 y(of)i(unit)h(v)m(ectors)f(tangen)m(t)h (to)f(\006)1219 2147 y Fn(i)1285 2132 y Fu(is)g(an)g(in)m(v)-6 b(arian)m(t)34 b(subset)g(for)g(the)f(geo)s(desic)h(\015o)m(w)f(on)h Fo(T)3380 2096 y Fk(1)3425 2132 y Fo(S)3492 2096 y Fk(3)3536 2132 y Fu(.)120 2252 y(Observ)m(e)39 b(that)f(eac)m(h)h Fo(O)1033 2267 y Fn(i)1104 2252 y Fu(is)f(h)m(yp)s(erb)s(olic)g(as)g (an)h(orbit)e(of)i(the)f(geo)s(desic)h(\015o)m(ws)f(of)h Fj(b)-5 b(oth)37 b Fu(\006)3566 2267 y Fn(i)p Fi(\000)p Fk(1)3739 2252 y Fu(and)0 2371 y(\006)72 2386 y Fn(i)105 2371 y Fu(.)48 b(It)34 b(follo)m(ws)g(that)g(the)h(deriv)-6 b(ativ)m(e)34 b(\(at)g(the)g(\014xed)h(p)s(oin)m(t)g(corresp)s(onding)g (to)f(the)h(closed)g(orbit\))f(of)0 2491 y(the)29 b(P)m(oincar)m(\023) -47 b(e)29 b(map)f(for)h Fo(O)1012 2506 y Fn(i)1073 2491 y Fu(has)h(t)m(w)m(o)e(expanding)h(eigen)m(v)m(ectors)h(and)f(t)m(w)m (o)g(con)m(tracting)f(eigen)m(v)m(ectors.)0 2610 y(Hence)34 b(eac)m(h)g Fo(O)596 2625 y Fn(i)663 2610 y Fu(is)f(a)g(h)m(yp)s(erb)s (olic)g(closed)h(orbit)f(for)g(the)h(geo)s(desic)g(\015o)m(w)f(of)h (the)f(new)h(metric)e(on)i Fo(S)3799 2574 y Fk(3)3843 2610 y Fu(.)120 2730 y(No)m(w)39 b(w)m(e)h(can)f(apply)h(Prop)s (osition)e(2.4)h(to)g(the)h(hetero)s(clinic)f(orbits)h(that)f(connect)h Fo(O)3489 2745 y Fn(i)3562 2730 y Fu(to)f Fo(O)3766 2745 y Fn(i)p Fk(+1)0 2849 y Fu(for)j(eac)m(h)h Fo(i)f Fm(2)h Fc(Z)p Fo(=m)p Fu(.)67 b(W)-8 b(e)42 b(can)g(arrange)g(that)g(these)h (new)f(p)s(erturbations)g(also)g(tak)m(e)f(place)h(inside)0 2969 y(the)d Fo(B)253 2984 y Fn(i)286 2969 y Fu('s)g(and)g(ha)m(v)m(e)g (supp)s(orts)h(disjoin)m(t)f(from)f(an)m(y)h(of)g(the)g(previous)g(p)s (erturbations.)61 b(After)39 b(these)0 3088 y(p)s(erturbations,)53 b(eac)m(h)d(of)f(the)g(orbits)g Fo(O)1602 3103 y Fn(i)1684 3088 y Fu(is)g(h)m(yp)s(erb)s(olic)g(and)h Fo(W)2621 3052 y Fn(u)2673 3088 y Fu(\()p Fo(O)2788 3103 y Fn(i)2821 3088 y Fu(\))e(and)i Fo(W)3226 3052 y Fn(s)3268 3088 y Fu(\()p Fo(O)3383 3103 y Fn(i)p Fk(+1)3517 3088 y Fu(\))f(ha)m(v)m(e) g(a)0 3208 y(transv)m(erse)39 b(in)m(tersection)h(for)f(eac)m(h)g Fo(i)e Fm(2)g Fc(Z)p Fo(=)p Fu(3)p Fo(m)p Fu(.)58 b(It)38 b(follo)m(ws)g(immediately)e(from)i(Smale's)f(theorem)0 3328 y(\(see)h Fm(x)p Fu(1\))f(that)f(there)i(is)f(a)g(horsesho)s(e)h (con)m(taining)g Fo(O)2064 3343 y Fk(1)2108 3328 y Fo(;)17 b(:)g(:)g(:)d(;)j(O)2406 3343 y Fk(3)p Fn(m)2521 3328 y Fu(.)56 b(Since)38 b(these)g(orbits)f(include)h(the)0 3447 y(orbits)i(with)g(whic)m(h)h(w)m(e)g(b)s(egan)g(the)f(pro)s(of,)i (the)f(horsesho)s(e)h(that)e(w)m(e)g(ha)m(v)m(e)h(created)g(is)f Fo(")p Fu(-dense)j(in)0 3567 y Fo(T)72 3530 y Fk(1)117 3567 y Fo(S)184 3530 y Fk(3)228 3567 y Fu(,)29 b(in)g(the)g(sense)g (explained)g(at)f(the)h(b)s(eginning)g(of)f(Section)h(2.)43 b(Theorem)28 b(0.1)f(no)m(w)i(follo)m(ws,)g(since)0 3686 y(the)34 b(restriction)f(of)g(the)h(geo)s(desic)f(\015o)m(w)h(to)f(the) g(horsesho)s(e)i(is)e(top)s(ologically)e(transitiv)m(e.)419 b Fg(\004)187 3906 y Fw(4.)55 b(T)-10 b(op)s(ological)37 b(transitivit)m(y)e(except)g(on)i(a)g(set)f(of)h(arbitrarily)e(small)i (measure)120 4085 y Fu(In)24 b(this)h(section)g(w)m(e)g(pro)m(v)m(e)g (Theorem)f(0.2.)40 b(The)25 b(pro)s(of)g(is)g(similar)d(to)i(that)g(of) h(Theorem)f(0.1.)41 b(Again)0 4204 y(w)m(e)31 b(create)f(a)h(sequence)h (of)e(orthogonal)g(spheres)i(and)f(apply)f(Prop)s(osition)f(2.2)h(to)f (create)i(a)f(sequence)0 4324 y(of)25 b(h)m(yp)s(erb)s(olic)f(closed)h (orbits)f(link)m(ed)h(b)m(y)f(hetero)s(clinic)h(orbits.)41 b(Then)25 b(w)m(e)g(will)e(use)i(Prop)s(osition)f(2.5)f(to)0 4444 y(join)k(all)g(of)g(these)h(hetero)s(clinic)f(orbits)g(in)m(to)g (a)g(single)g(orbit,)h(rather)f(than)g(applying)g(Prop)s(osition)f (2.4.)120 4563 y(The)42 b(ma)6 b(jor)40 b(di\013erence)j(from)e (Theorem)h(0.1,)g(ho)m(w)m(ev)m(er,)j(is)c(that)h(the)g(sequence)h(of)f (orthogonal)0 4683 y(spheres)d(is)d(in\014nite)h(rather)g(than)g (cyclic.)55 b(When)38 b(there)f(are)g(only)f(\014nitely)g(man)m(y)g (spheres,)j(w)m(e)f(can)0 4802 y(c)m(ho)s(ose)29 b(the)g(small)f(balls) g(in)h(whic)m(h)g(to)f(p)s(erturb)i(the)f(metric,)f Fj(after)g Fu(w)m(e)h(ha)m(v)m(e)g(c)m(hosen)h(the)f(sequence)i(of)0 4922 y(spheres,)e(simply)24 b(b)m(y)i(ensuring)h(that)e(the)i(sequence) g(of)f(spheres)i(is)d(in)h(general)g(p)s(osition)f(and)i(c)m(ho)s (osing)0 5041 y(the)h(balls)g(small)f(enough)i(so)f(that)g(eac)m(h)h (ball)e(in)m(tersects)j(only)d(the)h(there)h(spheres)h(that)e(pass)h (through)0 5161 y(its)k(cen)m(ter.)45 b(No)m(w)33 b(w)m(e)g(m)m(ust)g (c)m(ho)s(ose)h(the)f(spheres)i(and)f(balls)e(in)i(batc)m(hes.)45 b(Eac)m(h)33 b(ball)g(will)f(con)m(tain)h(a)0 5280 y(smaller)28 b(ball,)h(whic)m(h)h(w)m(e)g(call)e(its)h(core,)h(and)g(eac)m(h)g(core) g(will)e(con)m(tain)h(a)g(still)f(smaller)g(ball,)i(whic)m(h)g(w)m(e)0 5400 y(call)35 b(the)g(inner)h(core.)51 b(The)35 b(p)s(erturbations)h (to)f(the)h(metric)e(will)g(ev)m(en)m(tually)h(b)s(e)h(made)e(in)i(the) f(inner)1908 5639 y Fs(13)p eop %%Page: 14 14 14 13 bop 0 100 a Fu(cores)38 b(of)g(the)g(balls.)57 b(W)-8 b(e)39 b(ensure)g(that)e(the)h(spheres)i(in)d(eac)m(h)i(new)f (batc)m(h)h(do)f(not)f(in)m(tersect)i(of)f(the)0 219 y(inner)d(cores)g(of)f(the)h(previously)f(c)m(hosen)i(balls.)46 b(Then)36 b(w)m(e)e(p)s(erturb)h(the)g(new)g(spheres)h(so)e(that)g (they)0 339 y(are)g(in)h(general)f(p)s(osition)g(and)h(c)m(ho)s(ose)g (a)f(small)f(ball)h(around)h(one)f(the)h(in)m(tersection)g(p)s(oin)m (ts)f(of)h(eac)m(h)0 458 y(triple)e(of)g(consecutiv)m(e)i(spheres)g(in) e(the)h(batc)m(h.)120 578 y(W)-8 b(e)27 b(c)m(ho)s(ose)i(eac)m(h)f (batc)m(h)h(of)e(spheres)j(as)d(w)m(e)h(did)g(in)g(the)g(pro)s(of)f(of) h(Theorem)f(0.1,)h(b)m(y)g(\014rst)g(c)m(ho)s(osing)0 697 y(a)23 b(sequence)i(of)f(crosses)h(and)e(then)i(linking)d(the)i (crosses.)42 b(It)23 b(is)g(straigh)m(tforw)m(ard)h(to)f(c)m(ho)s(ose)h (the)g(crosses)0 817 y(so)36 b(that)g(they)g(miss)f(the)i(cores)g(of)f (the)g(pre-)h(existing)f(balls,)g(but)h(c)m(ho)s(osing)f(the)h(linking) e(spheres)j(so)0 936 y(that)32 b(they)g(miss)g(the)h(inner)g(cores)g (in)m(v)m(olv)m(es)f(a)h(subtlet)m(y)-8 b(.)43 b(The)33 b(spheres)i(whic)m(h)e(can)g(link)e(t)m(w)m(o)i(crosses)0 1056 y(\(\006)111 1013 y Fi(\000)111 1083 y Fk(1)178 1056 y Fo(;)17 b Fu(\006)295 1013 y Fk(+)295 1083 y(1)360 1056 y Fu(\))42 b(and)g(\(\006)754 1013 y Fi(\000)754 1083 y Fk(2)821 1056 y Fo(;)17 b Fu(\006)938 1013 y Fk(+)938 1083 y(2)1003 1056 y Fu(\))42 b(are)g(the)g(spheres)i(orthogonal)d(to)g (the)h(great)g(circle)g(\006)3220 1013 y Fk(+)3220 1083 y(1)3314 1056 y Fm(\\)28 b Fu(\006)3480 1013 y Fi(\000)3480 1083 y Fk(2)3547 1056 y Fu(.)70 b(These)0 1176 y(spheres)31 b(all)d(in)m(tersect)h(in)g(a)g(common)e(great)h(circle.)43 b(In)29 b(order)g(to)g(b)s(e)g(able)g(to)f(c)m(ho)s(ose)i(the)f(the)g (linking)0 1295 y(sphere)44 b(so)f(that)g(it)f(misses)h(the)g(inner)g (cores,)j(w)m(e)d(m)m(ust)g(ensure)h(that)e(this)h(common)e(great)i (circle)0 1415 y(misses)33 b(the)h(outer)f(cores)h(of)f(the)h (pre-existing)f(balls.)120 1585 y(W)-8 b(e)45 b(no)m(w)g(in)m(tro)s (duce)g(the)h(geometrical)d(to)s(ols)h(whic)m(h)h(enable)h(us)f(to)g(o) m(v)m(ercome)f(this)h(di\016cult)m(y)-8 b(.)0 1705 y(Although)34 b(these)h(ideas)g(come)e(from)h(pro)6 b(jectiv)m(e)34 b(geometry)-8 b(,)33 b(it)g(will)h(b)s(e)g(more)f(con)m(v)m(enien)m(t)j (for)f(us)f(to)0 1824 y(express)g(them)f(in)g(terms)g(of)g(the)h (geometry)e(of)h Fo(S)1896 1788 y Fk(3)1974 1824 y Fu(with)g(the)g (round)h(metric.)120 1944 y(The)41 b(in)m(tersection)g(of)g(the)g (sphere)h Fo(S)1559 1908 y Fk(3)1644 1944 y Fu(with)e(a)h Fo(k)s Fu(-dimensional)f(linear)g(subspace)j(of)e Fc(R)3507 1908 y Fk(4)3599 1944 y Fu(with)f(is)0 2063 y(a)f Fw(great)45 b Fu(\()p Fo(k)29 b Fm(\000)e Fu(1\))p Fw(-sphere)p Fu(.)62 b(The)40 b(case)h Fo(k)g Fu(=)e(0)g(giv)m(es)h(us)g(the)g(empt)m(y)f (set,)i(whic)m(h)g(has)f(top)s(ological)0 2183 y(dimension)35 b Fm(\000)p Fu(1)g(and)h(is)f(the)g(in)m(tersection)h(of)g Fo(S)1826 2147 y Fk(3)1905 2183 y Fu(with)f(the)g(space)i Fm(f)p Fu(0)p Fm(g)p Fu(.)49 b(The)35 b(cases)h Fo(k)e Fu(=)d(1)p Fo(;)17 b Fu(2)p Fo(;)g Fu(3)33 b(giv)m(e)0 2303 y(us)40 b(pairs)g(of)g(an)m(tip)s(o)s(dal)f(p)s(oin)m(ts,)j(great) d(circles)h(and)h(great)e(spheres)j(resp)s(ectiv)m(ely)-8 b(.)63 b(If)40 b Fo(\013)g Fu(is)g(a)f(great)0 2422 y Fo(k)s Fu(-sphere,)e(let)d Fo(V)632 2437 y Fn(\013)724 2422 y Fu(denote)i(the)f(corresp)s(onding)h(\()p Fo(k)26 b Fu(+)e(1\)-dimensional)34 b(subspace)j(of)e Fc(R)3351 2386 y Fk(4)3402 2422 y Fu(.)49 b(The)35 b Fw(dual)0 2542 y Fo(\013)64 2505 y Fi(?)166 2542 y Fu(of)f Fo(\013)g Fu(is)g(the)h(great)f(\(2)22 b Fm(\000)i Fo(k)s Fu(\)-sphere)35 b(corresp)s(onding)h(to)d(the)i(orthogonal)e(complemen)m(t)h(in)g Fc(R)3735 2505 y Fk(4)3820 2542 y Fu(of)0 2661 y Fo(V)58 2676 y Fn(\013)115 2661 y Fu(.)46 b(If)35 b Fo(\013)f Fu(and)g Fo(A)g Fu(are)g(great)g(spheres)h(with)f Fo(\013)29 b Fm(\032)h Fo(A)j Fu(and)i(dim)15 b Fo(\013)30 b Fm(\024)f Fo(k)j Fm(\024)d Fu(dim)16 b Fo(A)p Fu(,)34 b(w)m(e)g(de\014ne)i Fm(S)3590 2625 y Fn(k)3639 2661 y Fu(\()p Fo(\013;)17 b(A)p Fu(\))0 2781 y(to)36 b(b)s(e)i(the)f(collection)g(of)g(all)f (great)g Fo(k)s Fu(-spheres)j Fo(S)k Fu(with)37 b Fo(\013)d Fm(\022)g Fo(S)40 b Fm(\022)34 b Fo(A)p Fu(.)55 b(If)37 b Fo(\013)g Fu(is)g(a)g(great)g(circle,)g(then)0 2900 y Fm(S)67 2864 y Fk(2)112 2900 y Fu(\()p Fo(\013)215 2864 y Fi(?)283 2900 y Fo(;)17 b(S)395 2864 y Fk(3)438 2900 y Fu(\))30 b(is)h(the)f(space)i(of)e(all)g(great)g(2-spheres)i (that)e(in)m(tersect)h Fo(\013)f Fu(orthogonally)-8 b(.)41 b(If)31 b Fo(\013)d Fm(\032)g Fo(A)p Fu(,)i(then)0 3020 y Fo(A)75 2984 y Fi(?)170 3020 y Fm(\032)e Fo(\013)339 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Fo(\013;)17 b(A)p Fu(\).)0 3633 y(Since)34 b Fo(S)6 b(O)s Fu(\()p Fo(A)520 3597 y Fi(?)586 3633 y Fo(;)17 b(\013)695 3597 y Fi(?)762 3633 y Fu(\))27 b(=)i Fo(S)6 b(O)s Fu(\()p Fo(\013;)17 b(A)p Fu(\),)31 b(it)h(easy)i(to)e (pro)m(v)m(e)0 3804 y Fw(Lemma)42 b(4.1.)57 b Fj(The)38 b(bije)-5 b(ction)37 b Fo(S)h Fm(7!)32 b Fo(S)1552 3767 y Fi(?)1658 3804 y Fj(fr)-5 b(om)38 b Fm(S)1963 3767 y Fn(k)2013 3804 y Fu(\()p Fo(\013;)17 b(A)p Fu(\))37 b Fj(to)h Fm(S)2501 3767 y Fk(2)p Fi(\000)p Fn(k)2653 3804 y Fu(\()p Fo(A)2767 3767 y Fi(?)2834 3804 y Fo(;)17 b(\013)2943 3767 y Fi(?)3009 3804 y Fu(\))38 b Fj(c)-5 b(arries)37 b Fo(\026)3477 3824 y Fi(S)3531 3804 y Fb(k)3575 3824 y Fk(\()p Fn(\013;A)p Fk(\))3816 3804 y Fj(to)0 3923 y Fo(\026)60 3943 y Fi(S)114 3923 y Fb(k)q Fe(\000)p Fh(2)245 3943 y Fk(\()p Fn(A)336 3923 y Fe(?)395 3943 y Fn(;\013)471 3923 y Fe(?)530 3943 y Fk(\))566 3923 y Fj(.)120 4103 y Fu(No)m(w)43 b(supp)s(ose)i(that)e(w)m(e)h(ha)m(v)m (e)g(a)g(collection)f(of)h(balls)f(in)g 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Fk(0)96 4871 y Fj(,)41 b(then,)f(for)f(any)i (sp)-5 b(ac)g(e)39 b Fm(S)1115 4835 y Fn(k)1164 4871 y Fu(\()p Fo(\013;)17 b(A)p Fu(\))p Fj(,)39 b(wher)-5 b(e)40 b Fo(\013)g Fj(and)g Fo(A)f Fj(ar)-5 b(e)40 b(a)g(p)-5 b(air)39 b(of)g(gr)-5 b(e)g(at)40 b(spher)-5 b(es)38 b(with)i Fo(\013)35 b Fm(\032)h Fo(A)0 4991 y Fj(and)f Fo(k)k Fj(is)c(an)h(inte)-5 b(ger)34 b(with)i Fu(dim)15 b Fo(\013)28 b(<)g(k)j(<)d Fu(dim)16 b Fo(A)p Fj(,)35 b(we)g(have:)152 5136 y Fu(\(1\))50 b Fj(If)35 b Fo(\013)23 b Fm(\\)f(B)31 b Fu(=)e Fm(;)p Fj(,)35 b(then)g Fo(\026)1208 5156 y Fi(S)1262 5136 y Fb(k)1306 5156 y Fk(\()p Fn(\013;A)p Fk(\))1508 5136 y Fm(f)p Fo(S)f Fm(2)28 b(S)1814 5099 y Fn(k)1864 5136 y Fu(\()p Fo(\013;)17 b(A)p Fu(\))26 b(:)h Fo(S)h Fm(\\)23 b(C)34 b Fu(=)28 b Fm(;g)f Fo(>)h(:)p Fu(99)p Fj(.)152 5268 y Fu(\(2\))50 b Fj(If)35 b Fo(A)509 5232 y Fi(?)599 5268 y Fm(\\)22 b(B)31 b Fu(=)d Fm(;)p Fj(,)35 b(then)h Fo(\026)1286 5288 y Fi(S)1340 5268 y Fb(k)1384 5288 y Fk(\()p Fn(\013;A)p Fk(\))1586 5268 y Fm(f)p Fo(S)d Fm(2)c(S)1892 5232 y Fn(k)1941 5268 y Fu(\()p Fo(\013;)17 b(A)p Fu(\))27 b(:)g Fo(S)2352 5232 y Fi(?)2441 5268 y Fm(\\)c(C)34 b Fu(=)28 b Fm(;g)f Fo(>)h(:)p Fu(99)p Fj(.)152 5400 y Fu(\(3\))50 b Fj(If)35 b Fo(\013)23 b Fm(\\)f(C)34 b Fu(=)28 b Fm(;)p Fj(,)35 b(then)h Fo(\026)1198 5420 y Fi(S)1252 5400 y Fb(k)1295 5420 y Fk(\()p Fn(\013;A)p Fk(\))1498 5400 y Fm(f)p Fo(S)d Fm(2)28 b(S)1803 5364 y Fn(k)1853 5400 y Fu(\()p Fo(\013;)17 b(A)p Fu(\))27 b(:)g Fo(S)h Fm(\\)22 b(I)7 b(C)35 b Fu(=)28 b Fm(;g)f Fo(>)i(:)p Fu(99)p Fj(.)1908 5639 y Fs(14)p eop %%Page: 15 15 15 14 bop 152 100 a Fu(\(4\))50 b Fj(If)35 b Fo(A)509 63 y Fi(?)599 100 y Fm(\\)22 b(C)34 b Fu(=)28 b Fm(;)p Fj(,)35 b(then)h Fo(\026)1276 120 y Fi(S)1330 100 y Fb(k)1373 120 y Fk(\()p Fn(\013;A)p Fk(\))1576 100 y Fm(f)p Fo(S)d Fm(2)28 b(S)1881 63 y Fn(k)1931 100 y Fu(\()p Fo(\013;)17 b(A)p Fu(\))26 b(:)i Fo(S)2342 63 y Fi(?)2431 100 y Fm(\\)23 b(I)7 b(C)34 b Fu(=)28 b Fm(;g)g Fo(>)g(:)p Fu(99)p Fj(.)0 300 y(Pr)-5 b(o)g(of.)49 b Fu(W)-8 b(e)43 b(pro)m(v)m(e)g(\(1\))f(and)h (\(3\);)i(\(2\))d(and)h(\(4\))f(then)h(follo)m(w)f(easily)g(b)m(y)g (applying)g(\(1\))g(and)h(\(3\))e(to)0 420 y Fm(S)67 384 y Fk(2)p Fi(\000)p Fn(k)219 420 y Fu(\()p Fo(A)333 384 y Fi(?)400 420 y Fo(;)17 b(\013)509 384 y Fi(?)575 420 y Fu(\))32 b(and)g(using)g(Lemma)f(A.)g(Let)h Fo(\014)h Fu(=)28 b Fo(\013)2021 384 y Fi(?)2108 420 y Fm(\\)20 b Fo(A)31 b Fu(and)i(let)e Fo(B)37 b Fu(b)s(e)32 b(a)g(ball)f(of)h (radius)g Fo(r)j Fu(in)d Fo(S)3828 384 y Fk(3)3872 420 y Fu(.)0 539 y(Consider)j(the)g(radial)f(pro)6 b(jection)34 b(of)h Fo(A)23 b Fm(n)g(f)p Fo(\013)p Fm(g)34 b Fu(on)m(to)g Fo(\014)40 b Fu(that)34 b(is)h(de\014ned)h(b)m(y)f(mapping)e Fo(p)d Fm(2)g Fo(A)23 b Fm(n)g Fo(\013)34 b Fu(to)0 659 y(the)g(in)m(tersection)g(with)e Fo(\014)39 b Fu(of)34 b(the)f(unique)h(sphere)h(in)e Fm(S)2131 623 y Fk(1+dim)14 b Fn(\013)2435 659 y Fu(\()p Fo(\013;)j(A)p Fu(\))32 b(that)h(con)m(tains)h Fo(p)p Fu(.)120 779 y(If)g Fo(\013)22 b Fm(\\)h Fo(B)34 b Fu(=)29 b Fm(;)p Fu(,)k(then)i(this)e(pro)6 b(jection)34 b(maps)g(the)g(core)g Fo(C)41 b Fu(of)34 b(the)g(ball)g Fo(B)k Fu(to)33 b(the)i(union)f(of)g(a)f(ball)0 898 y(in)i Fo(\014)41 b Fu(with)35 b(radius)g(at)g(most)f Fo(r)k Fu(and)e(the)f(an)m(tip)s(o)s(dal)g(ball)f(in)i Fo(\014)5 b Fu(;)36 b(and)g(the)f(same)g(is)g(true)g(of)h(the)f(inner)0 1018 y(core)42 b(of)f Fo(B)46 b Fu(if)c Fo(\013)28 b Fm(\\)g Fo(C)48 b Fu(=)42 b Fm(;)p Fu(.)68 b(There)43 b(is)e(a)g(measure)h(preserving)g(bijection)g(b)s(et)m(w)m(een)h Fm(S)3388 982 y Fn(k)3437 1018 y Fu(\()p Fo(\013;)17 b(A)p Fu(\))40 b(and)0 1137 y Fm(S)67 1101 y Fn(k)r Fi(\000)p Fk(1)219 1137 y Fu(\()p Fm(;)p Fo(;)17 b(\014)5 b Fu(\).)43 b(The)34 b(prop)s(osition)f(follo)m(ws)g(from)f(the)i(next)f(lemma.) 1294 b Fg(\004)0 1309 y Fw(Lemma)45 b(4.3.)57 b Fj(Ther)-5 b(e)40 b(is)g(a)h(c)-5 b(onstant)40 b Fo(b)d(>)g Fu(0)j Fj(such)h(that,)h(if)d Fu(0)e Fm(\024)g Fo(j)43 b(<)37 b(l)h Fm(\024)g Fu(3)i Fj(and)g Fo(S)47 b Fj(is)40 b(a)h(gr)-5 b(e)g(at)40 b Fo(l)r Fj(-)0 1429 y(spher)-5 b(e,)32 b(then)g(the)g(pr) -5 b(ob)g(ability)32 b(\(with)f(r)-5 b(esp)g(e)g(ct)32 b(to)g Fo(\026)1943 1447 y Fi(S)1997 1427 y Fb(j)2034 1447 y Fk(\()p Fi(;)p Fn(;S)t Fk(\))2219 1429 y Fj(\))g(that)g(an)g (element)h(of)f Fm(S)3178 1392 y Fn(j)3221 1429 y Fu(\()p Fm(;)p Fo(;)17 b(S)6 b Fu(\))30 b Fj(interse)-5 b(cts)0 1550 y(a)36 b(given)e(b)-5 b(al)5 b(l)37 b(in)e Fo(S)41 b Fj(of)35 b(r)-5 b(adius)36 b Fo(r)30 b(<)e(\031)t(=)p Fu(1000)34 b Fj(is)h(at)h(most)g Fo(br)s Fj(.)0 1721 y(Pr)-5 b(o)g(of.)49 b Fu(W)-8 b(e)54 b(ma)m(y)d(assume)i(without)f (loss)h(of)h(generalit)m(y)e(that)g(the)i(ball)e(in)h(question)g(is)g (the)g Fo(r)s Fu(-)0 1841 y(neigh)m(b)s(ourho)s(o)s(d)c(in)f Fo(S)894 1805 y Fn(l)972 1841 y Fu(of)g(the)h(north)f(p)s(ole)g(\(0)p Fo(;)17 b(:)g(:)g(:)d(;)j Fu(0)p Fo(;)g Fu(1\).)86 b(Let)48 b Fo(E)58 b Fu(=)53 b Fo(S)2932 1805 y Fn(l)2994 1841 y Fm(\\)32 b Fu(\()p Fc(R)3197 1805 y Fn(l)3266 1841 y Fm(\002)g(f)p Fu(0)p Fm(g)p Fu(\))47 b(b)s(e)i(the)0 1960 y(equator.)43 b(The)34 b(probabilit)m(y)e(that)g(w)m(e)i(wish)f (to)f(estimate)g(is)h(the)g(same)f(as)h(the)g(probabilit)m(y)f(that)h (the)0 2080 y(\014rst)d Fo(j)22 b Fu(+)16 b(1)29 b(v)m(ectors)i(of)f(a) g(randomly)e(c)m(hosen)k(p)s(ositiv)m(ely)c(orien)m(ted)j(orthonormal)d (basis)j(of)f Fc(R)3533 2044 y Fn(l)p Fk(+)q(1)3700 2080 y Fu(span)0 2200 y(a)c(subspace)j(that)d(do)s(es)h(not)f(in)m(tersect)h (the)g(cone)g(subtended)i(at)d(the)h(origin)f(b)m(y)g(the)h(giv)m(en)g (ball.)41 b(If)27 b(the)0 2319 y(subspace)38 b(do)s(es)f(in)m(tersect)g (this)g(cone,)g(then)g(the)g(last)f(elemen)m(t)g(of)g(the)h(basis,)g (whic)m(h)h(is)e(orthogonal)0 2439 y(to)g(the)g(subspace,)i(m)m(ust)e (lie)f(in)i(the)f Fo(r)s Fu(-neigh)m(b)s(ourho)s(o)s(d)h Fo(N)2270 2454 y Fn(r)2350 2439 y Fu(of)f Fo(E)6 b Fu(.)52 b(The)36 b(probabilit)m(y)f(of)i(this)f(is)g(the)0 2558 y(measure)28 b(of)h Fo(N)572 2573 y Fn(r)644 2558 y Fu(with)f(resp)s (ect)i(to)e(the)g(Leb)s(esgue)i(measure)f(on)f Fo(S)2496 2522 y Fn(l)2555 2558 y Fu(\(normalized)f(to)h(b)s(e)h(a)f(probabilit)m (y)0 2678 y(measure\).)44 b(It)33 b(is)g(clear)g(the)h(v)m(olume)e(of)h Fo(N)1620 2693 y Fn(r)1698 2678 y Fu(is)g Fo(O)s Fu(\()p Fo(r)s Fu(\).)1792 b Fg(\004)120 2850 y Fu(Let)33 b(us)h(call)f(a)g (great)f(circle)i Fo(\033)j Fw(go)s(o)s(d)d Fu(if)f Fo(\033)26 b Fm(\\)c(B)31 b Fu(=)d Fm(;)33 b Fu(and)h Fo(\033)2377 2813 y Fi(?)2466 2850 y Fm(\\)23 b(B)31 b Fu(=)d Fm(;)p Fu(.)0 3021 y Fw(Prop)s(osition)43 b(4.4.)58 b Fj(Given)38 b Fo(")d(>)g Fu(0)p Fj(,)40 b(we)f(c)-5 b(an)40 b(cho)-5 b(ose)38 b Fo(\032)2178 3036 y Fn(")2255 3021 y Fo(>)d Fu(0)k Fj(such)g(that,)h(if)f(the)g(sum)h(of)f(the)g(r)-5 b(adii)0 3141 y(of)44 b(the)h(b)-5 b(al)5 b(ls)45 b(is)f(less)h(than)f Fo(\032)1135 3156 y Fn(")1177 3141 y Fj(,)j(then)e(the)f(set)h Fm(G)50 b Fj(of)45 b(go)-5 b(o)g(d)44 b(gr)-5 b(e)g(at)44 b(cir)-5 b(cles)44 b(is)g(a)h(subset)f(of)g Fm(S)3571 3105 y Fk(1)3617 3141 y Fu(\()p Fm(;)p Fo(;)17 b(S)3818 3105 y Fk(3)3861 3141 y Fu(\))0 3260 y Fj(satisfying)34 b Fo(\026)503 3278 y Fi(S)557 3259 y Fh(1)596 3278 y Fk(\()p Fi(;)p Fn(;S)745 3259 y Fh(3)783 3278 y Fk(\))819 3260 y Fu(\()p Fm(G)6 b Fu(\))28 b Fm(\025)g Fu(1)22 b Fm(\000)g Fo("=)p Fu(2)p Fj(.)0 3432 y(Pr)-5 b(o)g(of.)49 b Fu(This)34 b(follo)m(ws)f(from)f(Lemmas)g(4.1)h(and)g(4.3.)1815 b Fg(\004)0 3604 y Fw(Prop)s(osition)42 b(4.5.)57 b Fj(Assume)39 b(that)f(the)g(b)-5 b(al)5 b(ls)39 b(satisfy)e(\(1\){\(4\))g(fr)-5 b(om)38 b(Pr)-5 b(op)g(osition)37 b(4.2.)52 b(L)-5 b(et)39 b Fo(X)3746 3619 y Fk(1)3823 3604 y Fu(=)0 3725 y(\(\006)111 3682 y Fi(\000)111 3751 y Fk(1)178 3725 y Fo(;)17 b Fu(\006)295 3682 y Fk(+)295 3751 y(1)360 3725 y Fu(\))34 b Fj(b)-5 b(e)33 b(a)h(cr)-5 b(oss)33 b(such)h(that)g(the)g(dual)g(of)g Fu(\006)1880 3682 y Fk(+)1880 3751 y(1)1980 3725 y Fj(do)-5 b(es)33 b(not)h(interse)-5 b(ct)33 b(the)g(b)-5 b(al)5 b(ls.)46 b(L)-5 b(et)34 b Fo(\033)3428 3740 y Fk(2)3506 3725 y Fj(b)-5 b(e)34 b(a)g(go)-5 b(o)g(d)0 3847 y(gr)g(e)g(at)35 b(cir)-5 b(cle)35 b(that)h(makes)f(an)h(angle)f(gr)-5 b(e)g(ater)35 b(than)h Fo(\031)t(=)p Fu(4)e Fj(with)i Fu(\006)2457 3804 y Fk(+)2457 3874 y(1)2523 3847 y Fj(.)45 b(L)-5 b(et)36 b Fo(\033)2828 3862 y Fk(3)2908 3847 y Fj(b)-5 b(e)36 b(another)f(gr)-5 b(e)g(at)35 b(cir)-5 b(cle.)0 3969 y(Then)36 b(ther)-5 b(e)35 b(is)g(a)g(cr)-5 b(oss)35 b Fo(X)1035 3984 y Fk(2)1107 3969 y Fu(=)28 b(\(\006)1323 3927 y Fi(\000)1323 3996 y Fk(2)1390 3969 y Fo(;)17 b Fu(\006)1507 3927 y Fk(+)1507 3996 y(2)1573 3969 y Fu(\))35 b Fj(c)-5 b(enter)g(e)g(d)35 b(at)g Fo(\033)2209 3984 y Fk(2)2290 3969 y Fj(such)g(that:)152 4118 y Fu(\(1\))50 b(\006)402 4075 y Fi(\000)402 4144 y Fk(2)505 4118 y Fj(and)35 b Fu(\006)770 4075 y Fk(+)770 4144 y(2)872 4118 y Fj(do)g(not)h(me)-5 b(et)36 b(the)f(c)-5 b(or)g(es)35 b(of)g(the)h(b)-5 b(al)5 b(ls.)152 4237 y Fu(\(2\))50 b Fo(X)413 4252 y Fk(1)489 4237 y Fj(and)32 b Fo(X)762 4252 y Fk(2)838 4237 y Fj(c)-5 b(an)32 b(b)-5 b(e)31 b(linke)-5 b(d)32 b(by)g(a)g(gr)-5 b(e)g(at)31 b(spher)-5 b(e)32 b Fu(\006)f Fj(that)h(do)-5 b(es)32 b(not)g(interse)-5 b(ct)31 b(the)g(inner)h(c)-5 b(or)g(es)330 4357 y(of)35 b(the)g(b)-5 b(al)5 b(ls.)152 4477 y Fu(\(3\))50 b Fj(The)36 b(duals)g(of)f Fu(\006)980 4434 y Fi(\000)980 4503 y Fk(2)1083 4477 y Fj(and)g Fu(\006)1348 4434 y Fk(+)1348 4503 y(2)1450 4477 y Fj(do)g(not)h(interse)-5 b(ct)35 b(the)g(b)-5 b(al)5 b(ls.)152 4599 y Fu(\(4\))50 b Fj(The)36 b(angle)f(b)-5 b(etwe)g(en)35 b Fu(\006)1229 4556 y Fk(+)1229 4625 y(2)1331 4599 y Fj(and)h Fo(\033)1582 4614 y Fk(3)1662 4599 y Fj(is)f(at)g(le)-5 b(ast)36 b Fo(\031)t(=)p Fu(4)p Fj(.)0 4800 y(Pr)-5 b(o)g(of.)49 b Fu(Since)32 b Fo(X)654 4815 y Fk(2)730 4800 y Fu(is)f(cen)m(tered)i(at)d(the)i(go)s(o)s(d)e (great)h(circle)g Fo(\033)t Fu(,)g(the)g(duals)h(of)f(\006)2994 4757 y Fi(\000)2994 4826 y Fk(2)3092 4800 y Fu(and)h(\006)3356 4757 y Fk(+)3356 4826 y(2)3453 4800 y Fu(b)s(oth)f(lie)g(in)0 4921 y Fo(\033)61 4885 y Fi(?)57 4945 y Fk(2)128 4921 y Fu(,)i(whic)m(h)h(misses)f(the)g(balls)g(b)s(ecause)i Fo(\033)1610 4936 y Fk(2)1688 4921 y Fu(is)e(go)s(o)s(d.)43 b(Th)m(us)35 b(\(3\))d(will)g(hold)i(for)f(an)m(y)g(c)m(hoice)i(of)e Fo(X)3726 4936 y Fk(2)3770 4921 y Fu(.)120 5041 y(The)i(cross)i Fo(X)657 5056 y Fk(2)736 5041 y Fu(is)e(uniquely)h(determined)f(b)m(y)h (\006)1980 4999 y Fi(\000)1980 5068 y Fk(2)2047 5041 y Fu(.)50 b(W)-8 b(e)36 b(can)g(think)f(of)h(the)f(c)m(hoice)h(of)g (\006)3524 4999 y Fi(\000)3524 5068 y Fk(2)3627 5041 y Fu(in)f(t)m(w)m(o)0 5161 y(di\013eren)m(t)f(w)m(a)m(ys.)120 5280 y(On)c(the)h(one)f(hand,)h(\006)976 5238 y Fi(\000)976 5307 y Fk(2)1071 5280 y Fm(2)d(S)1232 5244 y Fk(2)1278 5280 y Fu(\()p Fo(\033)1374 5295 y Fk(2)1418 5280 y Fo(;)17 b(S)1530 5244 y Fk(3)1574 5280 y Fu(\).)43 b(Since)30 b Fo(\033)1997 5295 y Fk(2)2072 5280 y Fu(is)g(go)s(o)s(d,)g(it)f(do)s (es)h(not)g(meet)g(the)g(balls,)g(and)h(w)m(e)0 5400 y(can)36 b(apply)g(\(1\))f(of)h(Prop)s(osition)e(4.2)h(to)h(sho)m(w)g (that)f(the)h(set)g Fm(G)2425 5415 y Fk(1)2506 5400 y Fu(of)g(spheres)i(in)d Fm(S)3160 5364 y Fk(2)3206 5400 y Fu(\()p Fo(\033)3302 5415 y Fk(2)3346 5400 y Fo(;)17 b(S)3458 5364 y Fk(3)3502 5400 y Fu(\))35 b(that)h(do)1908 5639 y Fs(15)p eop %%Page: 16 16 16 15 bop 0 100 a Fu(not)37 b(meet)g(the)h(cores)g(of)f(the)h(balls)f (satis\014es)h Fo(\026)1820 118 y Fi(S)1874 98 y Fh(2)1913 118 y Fk(\()p Fn(\033)1990 128 y Fh(2)2030 118 y Fn(;S)2107 98 y Fh(3)2145 118 y Fk(\))2181 100 y Fu(\()p Fm(G)2279 115 y Fk(1)2324 100 y Fu(\))c Fm(\025)h Fo(:)p Fu(99.)56 b(Rotation)37 b(b)m(y)g Fo(\031)t(=)p Fu(2)f(ab)s(out)i Fo(\033)3828 115 y Fk(2)3872 100 y Fu(,)0 231 y(whic)m(h)d(mo)m(v)m(es) f(a)g(c)m(hoice)h(for)f(\006)1190 188 y Fi(\000)1190 257 y Fk(2)1292 231 y Fu(to)f(the)i(corresp)s(onding)g(c)m(hoice)g(for) g(\006)2751 188 y Fk(+)2751 257 y(2)2817 231 y Fu(,)f(is)g(a)g(measure) g(preserving)0 353 y(map)e(of)i Fm(S)402 317 y Fk(2)447 353 y Fu(\()p Fo(\033)543 368 y Fk(2)588 353 y Fo(;)17 b(S)700 317 y Fk(3)743 353 y Fu(\).)44 b(Hence)34 b(the)g(set)f Fm(G)1536 368 y Fk(2)1609 353 y Fm(\032)28 b(G)1773 368 y Fk(1)1851 353 y Fu(consisting)34 b(of)f(c)m(hoices)h(for)g(\006)2985 311 y Fi(\000)2985 380 y Fk(2)3085 353 y Fu(suc)m(h)h(that)e(b)s(oth)g (\006)3833 311 y Fi(\000)3833 380 y Fk(2)0 475 y Fu(and)j(the)g (corresp)s(onding)h(\006)1085 433 y Fk(+)1085 502 y(2)1186 475 y Fu(miss)e(the)h(cores)g(of)g(the)g(balls)g(satis\014es)g Fo(\026)2792 493 y Fi(S)2846 474 y Fh(2)2885 493 y Fk(\()p Fn(\033)2962 503 y Fh(2)3001 493 y Fn(;S)3078 474 y Fh(3)3116 493 y Fk(\))3152 475 y Fu(\()p Fm(G)3250 490 y Fk(2)3295 475 y Fu(\))31 b Fm(\025)h Fo(:)p Fu(98.)51 b(Th)m(us)0 607 y(most)32 b Fo(X)327 622 y Fk(2)405 607 y Fu(satisfy)h(\(1\):)43 b(more)32 b(precisely)i Fo(X)1655 622 y Fk(2)1732 607 y Fu(satis\014es)g(\(1\))f(if)g(\006)2423 564 y Fi(\000)2423 633 y Fk(2)2518 607 y Fm(2)28 b(G)2671 622 y Fk(2)2716 607 y Fu(.)120 726 y(It)h(is)g(not)h(di\016cult)g(to)f(sho)m(w)h(that)f (the)h(set)g(of)g(spheres)i(in)d Fm(S)2385 690 y Fk(2)2431 726 y Fu(\()p Fo(\033)2527 741 y Fk(2)2571 726 y Fo(;)17 b(S)2683 690 y Fk(3)2727 726 y Fu(\))29 b(that)h(mak)m(e)e(angle)i (less)g(than)0 847 y Fo(\031)t(=)p Fu(4)k(with)g Fo(\033)480 862 y Fk(3)560 847 y Fu(has)h(measure)g(at)f(most)f(1)p Fo(=)p Fu(2.)48 b(Hence)36 b(the)f(set)g Fm(G)2410 862 y Fk(3)2485 847 y Fm(\032)c(G)2652 862 y Fk(2)2732 847 y Fu(consisting)k(of)g(c)m(hoices)h(of)f(\006)3833 804 y Fi(\000)3833 873 y Fk(2)0 966 y Fu(suc)m(h)g(that)e(b)s(oth)g(\(1\))f (and)i(\(4\))f(hold)g(satis\014es)h Fo(\026)1839 984 y Fi(S)1893 964 y Fh(2)1932 984 y Fk(\()p Fn(\033)2009 994 y Fh(2)2048 984 y Fn(;S)2125 964 y Fh(3)2163 984 y Fk(\))2199 966 y Fu(\()p Fm(G)2297 981 y Fk(3)2342 966 y Fu(\))27 b Fm(\025)i Fo(:)p Fu(48.)120 1098 y(On)k(the)h(other)f (hand,)g(\006)1065 1055 y Fi(\000)1065 1124 y Fk(2)1165 1098 y Fu(is)g(uniquely)g(determined)g(b)m(y)g(the)h(great)e(circle)h (\006)3085 1055 y Fi(\000)3085 1124 y Fk(2)3174 1098 y Fm(\\)22 b Fu(\006)3334 1055 y Fk(+)3334 1124 y(1)3433 1098 y Fu(in)33 b(whic)m(h)h(it)0 1220 y(in)m(tersects)27 b(\006+)583 1235 y Fk(1)628 1220 y Fu(.)42 b(Let)26 b Fo(q)32 b Fu(=)c Fm(f)p Fo(q)1148 1184 y Fi(0)1176 1220 y Fo(;)17 b(q)1269 1184 y Fi(0)o(0)1319 1220 y Fm(g)28 b Fu(=)g Fo(\033)1559 1235 y Fk(2)1611 1220 y Fm(\\)8 b Fu(\006)1757 1177 y Fk(+)1757 1246 y(1)1823 1220 y Fu(.)42 b(Then)27 b(\006)2218 1177 y Fi(\000)2218 1246 y Fk(2)2293 1220 y Fm(\\)8 b Fu(\006)2439 1177 y Fk(+)2439 1246 y(1)2531 1220 y Fu(lies)26 b(in)g Fm(S)2872 1184 y Fk(1)2917 1220 y Fu(\()p Fo(q)t(;)17 b Fu(\006)3121 1177 y Fk(+)3121 1246 y(1)3186 1220 y Fu(\).)42 b(Since)26 b(the)h(dual)0 1342 y(of)k(\006)183 1300 y Fk(+)183 1369 y(1)279 1342 y Fu(do)s(es)g(not)g(meet)e(the)i(balls,)g(w)m(e)f(can)h (apply)g(\(2\))e(of)i(Prop)s(osition)f(4.2)f(to)h(sho)m(w)h(that)f(the) h(set)g Fm(G)3855 1357 y Fk(4)0 1465 y Fu(of)37 b Fo(\033)h Fm(2)c(S)379 1428 y Fk(1)425 1465 y Fu(\()p Fo(q)t(;)17 b Fu(\006)629 1422 y Fk(+)629 1491 y(1)694 1465 y Fu(\))37 b(suc)m(h)h(that)f Fo(\033)1279 1428 y Fi(?)1383 1465 y Fu(misses)g(the)g(cores)h(of)f(the)g(balls)g(satis\014es)h Fo(\026)3082 1496 y Fi(S)3136 1476 y Fh(1)3175 1496 y Fk(\()p Fn(q)s(;)p Fk(\006)3327 1463 y Fh(+)3327 1517 y(1)3382 1496 y Fk(\))3418 1465 y Fu(\()p Fm(G)3516 1480 y Fk(4)3561 1465 y Fu(\))33 b Fm(\025)i Fo(:)p Fu(99.)0 1612 y(By)c(\(3\))h(of)g(Prop)s(osition)g(4.2,)f(w)m(e)i(see)g(that)f (for)g(an)m(y)h Fo(\033)e Fm(2)d(G)2251 1627 y Fk(4)2328 1612 y Fu(there)33 b(are)f(spheres)i(in)f Fm(S)3279 1576 y Fk(2)3324 1612 y Fu(\()p Fo(\033)3424 1576 y Fi(?)3491 1612 y Fo(;)17 b(S)3603 1576 y Fk(3)3646 1612 y Fu(\))32 b(that)0 1731 y(miss)i(the)g(inner)h(cores)g(of)g(the)f(balls.)47 b(But)34 b Fm(S)1731 1695 y Fk(2)1777 1731 y Fu(\()p Fo(\033)1877 1695 y Fi(?)1943 1731 y Fo(;)17 b(S)2055 1695 y Fk(3)2099 1731 y Fu(\))34 b(is)g(the)h(space)g(of)g(great)f (spheres)i(orthogonal)0 1851 y(to)e Fo(\033)t Fu(;)f(an)m(y)h(of)g (these)h(spheres)h(can)e(b)s(e)h(used)g(to)e(link)h Fo(X)2106 1866 y Fk(1)2184 1851 y Fu(to)f(the)i(cross)f Fo(X)2807 1866 y Fk(2)2886 1851 y Fu(determined)g(b)m(y)g Fo(\033)t Fu(.)46 b(Th)m(us)0 1971 y(most)32 b Fo(X)327 1986 y Fk(2)405 1971 y Fu(satisfy)h(\(2\):)43 b(more)32 b(precisely)i Fo(X)1655 1986 y Fk(2)1732 1971 y Fu(satis\014es)g(\(2\))f(if)g(\006) 2423 1928 y Fi(\000)2423 1997 y Fk(2)2512 1971 y Fm(\\)23 b Fu(\006)2673 1928 y Fk(+)2673 1997 y(1)2767 1971 y Fm(2)28 b(G)2920 1986 y Fk(4)2965 1971 y Fu(.)120 2090 y(It)22 b(remains)h(to)f(sho)m(w)i(that)e(the)i(t)m(w)m(o)f(notions)g (of)g(\\most")e(in)i(the)h(ab)s(o)m(v)m(e)f(statemen)m(ts)f(are)i (su\016cien)m(tly)0 2210 y(compatible)33 b(to)g(enable)h(us)h(to)e(c)m (ho)s(ose)i(\006)1582 2167 y Fi(\000)1582 2236 y Fk(2)1682 2210 y Fu(so)f(that)g(\006)2094 2167 y Fi(\000)2094 2236 y Fk(2)2189 2210 y Fm(2)29 b(G)2343 2225 y Fk(3)2422 2210 y Fu(and)34 b(\006)2688 2167 y Fi(\000)2688 2236 y Fk(2)2778 2210 y Fm(\\)23 b Fu(\006)2939 2167 y Fk(+)2939 2236 y(1)3033 2210 y Fm(2)29 b(G)3187 2225 y Fk(4)3232 2210 y Fu(.)46 b(Th)m(us)35 b(w)m(e)f(wish)0 2332 y(to)39 b(sho)m(w)h(that)f Fo(\036)662 2296 y Fi(\000)p Fk(1)769 2332 y Fu(\()p Fm(G)867 2347 y Fk(4)912 2332 y Fu(\))25 b Fm(\\)i(G)1128 2347 y Fk(3)1211 2332 y Fm(6)p Fu(=)38 b Fm(;)p Fu(,)i(where)g Fo(\036)e Fu(:)g Fm(S)1968 2296 y Fk(2)2013 2332 y Fu(\()p Fo(\033)2109 2347 y Fk(2)2154 2332 y Fo(;)17 b(S)2266 2296 y Fk(3)2309 2332 y Fu(\))38 b Fm(!)f(S)2590 2296 y Fk(1)2636 2332 y Fu(\()p Fo(q)t(;)17 b Fu(\006)2840 2289 y Fk(+)2840 2359 y(1)2905 2332 y Fu(\))39 b(tak)m(es)g Fo(S)k Fm(2)c(S)3520 2296 y Fk(2)3565 2332 y Fu(\()p Fo(\033)3661 2347 y Fk(2)3705 2332 y Fo(;)17 b(S)3817 2296 y Fk(3)3861 2332 y Fu(\))0 2454 y(to)40 b Fo(S)32 b Fm(\\)27 b Fu(\006)387 2412 y Fk(+)387 2481 y(1)453 2454 y Fu(.)64 b(In)41 b(order)f(to)g(this,)h(w)m(e)g(need)g (to)e(estimate)g(the)i(Jacobian)f(of)g Fo(\036)g Fu(with)g(resp)s(ect)h (to)f(the)0 2574 y(measures)34 b Fo(\026)488 2592 y Fi(S)542 2572 y Fh(2)580 2592 y Fk(\()p Fn(\033)657 2602 y Fh(2)697 2592 y Fn(;S)774 2572 y Fh(3)812 2592 y Fk(\))881 2574 y Fu(and)g Fo(\026)1135 2605 y Fi(S)1189 2585 y Fh(1)1228 2605 y Fk(\()p Fn(q)s(;)p Fk(\006)1380 2573 y Fh(+)1380 2627 y(1)1435 2605 y Fk(\))1471 2574 y Fu(.)120 2718 y(Observ)m(e)25 b(that)f(there)h(is)g(a)f(natural)g(iden)m (ti\014cation)h(of)f Fm(S)2215 2682 y Fk(2)2261 2718 y Fu(\()p Fo(\033)2357 2733 y Fk(2)2401 2718 y Fo(;)17 b(S)2513 2682 y Fk(3)2557 2718 y Fu(\))24 b(with)g(the)h(set)g(of)f(an) m(tip)s(o)s(dal)g(pairs)0 2837 y(in)38 b(the)g(circle)h Fo(C)634 2852 y Fk(1)717 2837 y Fu(of)f(unit)g(v)m(ectors)g(at)g Fo(q)1568 2801 y Fi(0)1634 2837 y Fu(that)g(are)g(orthogonal)f(to)h Fo(\033)2713 2852 y Fk(2)2757 2837 y Fu(.)59 b(With)38 b(this)g(iden)m(ti\014cation,)0 2957 y Fo(\026)60 2975 y Fi(S)114 2955 y Fh(2)153 2975 y Fk(\()p Fn(\033)230 2985 y Fh(2)269 2975 y Fn(;S)346 2955 y Fh(3)384 2975 y Fk(\))447 2957 y Fu(is)27 b(the)h(measure)f(induced)i(on)e(the)g(set) g(of)h(an)m(tip)s(o)s(dal)e(pairs)h(b)m(y)g(Leb)s(esgue)i(measure)e(on) g Fo(C)3827 2972 y Fk(1)3872 2957 y Fu(.)0 3088 y(Similarly)e(there)j (is)f(a)g(natural)g(iden)m(ti\014cation)g(of)h Fm(S)1950 3052 y Fk(1)1995 3088 y Fu(\()p Fo(q)t(;)17 b Fu(\006)2199 3045 y Fk(+)2199 3115 y(1)2264 3088 y Fu(\))27 b(with)g(the)h(set)f(of) h(an)m(tip)s(o)s(dal)e(pairs)h(in)h(the)0 3210 y(circle)i Fo(C)328 3225 y Fk(2)404 3210 y Fu(of)g(unit)h(v)m(ectors)f(tangen)m(t) h(to)e(\006)1607 3168 y Fk(+)1607 3237 y(1)1704 3210 y Fu(at)h Fo(q)1871 3174 y Fi(0)1899 3210 y Fu(.)43 b(The)30 b(measure)h Fo(\026)2617 3242 y Fi(S)2671 3222 y Fh(1)2710 3242 y Fk(\()p Fn(q)s(;)p Fk(\006)2862 3209 y Fh(+)2862 3263 y(1)2917 3242 y Fk(\))2983 3210 y Fu(b)s(ecomes)f(the)h(measure)0 3342 y(induced)k(on)e(the)h(space)g(of)f(an)m(tip)s(o)s(dal)g(pairs)g (b)m(y)h(Leb)s(esgue)h(measure)e(on)g Fo(C)2917 3357 y Fk(2)2962 3342 y Fu(.)120 3462 y(The)j(Jacobian)f(of)h Fo(\036)g Fu(that)f(w)m(e)h(wish)g(to)f(estimate)f(is)i(the)g(same)f (as)g(the)h(Jacobian)g(\(with)f(resp)s(ect)0 3595 y(to)42 b(the)h(Leb)s(esgue)h(measures\))f(of)g(the)g(map)1773 3569 y Ff(b)1763 3595 y Fo(\036)g Fu(:)g Fo(C)2007 3610 y Fk(1)2095 3595 y Fm(!)g Fo(C)2309 3610 y Fk(2)2397 3595 y Fu(de\014ned)h(b)m(y)f(\014rst)g(pro)6 b(jecting)43 b(v)m(ectors)0 3716 y(in)h Fo(C)198 3731 y Fk(1)287 3716 y Fu(to)f(the)h(plane)g Fo(T)935 3731 y Fn(q)975 3711 y Fe(0)1006 3716 y Fu(\006)1078 3673 y Fk(+)1078 3742 y(1)1188 3716 y Fu(along)g(the)g(lines)g(in)g Fo(T)2070 3731 y Fn(q)2110 3711 y Fe(0)2141 3716 y Fo(S)2208 3680 y Fk(3)2296 3716 y Fu(that)g(are)g(parallel)f(to)g Fo(T)3255 3731 y Fn(q)3295 3711 y Fe(0)3326 3716 y Fo(\033)3383 3731 y Fk(2)3428 3716 y Fu(,)j(and)e(then)0 3835 y(normalizing)36 b(the)h(resulting)h(v)m(ectors.)56 b(An)37 b(elemen)m(tary)f (calculation)h(sho)m(ws)h(that)f(the)g(Jacobian)h(of)10 3937 y Ff(b)0 3963 y Fo(\036)h Fu(lies)g(b)s(et)m(w)m(een)h(sin)17 b Fo(\013)39 b Fu(and)g(csc)18 b Fo(\013)p Fu(,)40 b(where)f Fo(\013)g Fu(is)g(the)g(angle)g(b)s(et)m(w)m(een)h Fo(\033)2776 3978 y Fk(2)2859 3963 y Fu(and)g(\006)3131 3920 y Fk(+)3131 3989 y(1)3197 3963 y Fu(.)61 b(Since)39 b(w)m(e)h(ha)m(v)m(e)0 4090 y(assumed)31 b(that)f Fo(\013)e Fm(\025)g Fo(\031)t(=)p Fu(4,)i(it)g(follo)m(ws)g(that)g(the)h(Jacobian)g(of)g Fo(\036)g Fu(lies)f(b)s(et)m(w)m(een)i(1)p Fo(=)3095 4007 y Fm(p)p 3178 4007 50 4 v 83 x Fu(2)e(and)3449 4007 y Fm(p)p 3532 4007 V 83 x Fu(2.)43 b(Using)0 4209 y(this)29 b(and)h(our)g(earlier)f(estimates,)g(w)m(e)h(see)g(that)f Fo(\026)1890 4227 y Fi(S)1944 4207 y Fh(2)1983 4227 y Fk(\()p Fn(\033)2060 4237 y Fh(2)2099 4227 y Fn(;S)2176 4207 y Fh(3)2214 4227 y Fk(\))2250 4209 y Fu(\()p Fo(\036)2348 4173 y Fi(\000)p Fk(1)2455 4209 y Fu(\()p Fm(G)2553 4224 y Fk(4)2598 4209 y Fu(\)\))14 b(+)g Fo(\026)2841 4227 y Fi(S)2895 4207 y Fh(2)2935 4227 y Fk(\()p Fn(\033)3012 4237 y Fh(2)3051 4227 y Fn(;S)3128 4207 y Fh(3)3166 4227 y Fk(\))3202 4209 y Fu(\()p Fm(G)3300 4224 y Fk(3)3345 4209 y Fu(\))27 b Fo(>)i Fu(1.)42 b(Hence)0 4336 y Fo(\036)59 4300 y Fi(\000)p Fk(1)166 4336 y Fu(\()p Fm(G)264 4351 y Fk(4)309 4336 y Fu(\))22 b Fm(\\)g(G)517 4351 y Fk(3)590 4336 y Fm(6)p Fu(=)28 b Fm(;)p Fu(,)33 b(as)g(desired.)2563 b Fg(\004)0 4573 y Fw(Pro)s(of)48 b(of)f(Theorem)f(0.2.)57 b Fu(Let)41 b Fo(")f(>)h Fu(0)f(b)s(e)h(giv)m(en.)66 b(Cho)s(ose)40 b Fo(\032)g(>)g Fu(0)g(small)f(enough)j(so)e(that)h(the) 0 4692 y(conclusions)29 b(of)e(Prop)s(ositions)g(4.2,)h(4.4)e(and)i (4.5)f(hold)h(for)f(an)m(y)h(collection)f(of)g(balls)g(whose)i (diameters)0 4812 y(ha)m(v)m(e)34 b(sum)f(less)g(than)h Fo(\032)p Fu(.)43 b(W)-8 b(e)34 b(shall)f(construct)h(an)f(in\014nite)h (sequence)h(of)e(orthogonal)g(spheres)1140 5046 y(\006)1212 5005 y Fk(0)1212 5071 y(1)1257 5046 y Fo(;)17 b Fu(\006)1374 5005 y Fk(1)1374 5071 y(1)1418 5046 y Fo(;)g(:)g(:)g(:)d(;)j Fu(\006)1712 5005 y Fk(1)1712 5071 y Fn(m)1783 5081 y Fh(1)1825 5046 y Fo(;)g Fu(\006)1942 5005 y Fk(2)1942 5071 y(1)1986 5046 y Fo(;)g(:)g(:)g(:)e(;)i Fu(\006)2281 5005 y Fk(2)2281 5071 y Fn(m)2352 5081 y Fh(2)2394 5046 y Fo(;)g Fu(\006)2511 5005 y Fk(3)2511 5071 y(1)2555 5046 y Fo(;)g(:)g(:)g(:)d(:)0 5280 y Fu(The)29 b(sequence)h(will)e(b)s (e)h(nondegenerate,)i(so)d(all)g(of)h(its)f(terms)g(are)g(distinct,)i (an)m(y)e(three)h(spheres)i(from)0 5400 y(the)39 b(sequence)i(in)m (tersect)e(in)g(a)g(pair)f(of)h(an)m(tip)s(o)s(dal)f(p)s(oin)m(ts,)i (and)g(at)e(most)g(three)h(spheres)i(from)d(the)1908 5639 y Fs(16)p eop %%Page: 17 17 17 16 bop 0 100 a Fu(sequence)40 b(can)f(in)m(tersect)g(at)f(an)m(y)g (p)s(oin)m(t)g(of)h Fo(S)1784 63 y Fk(3)1828 100 y Fu(.)59 b(These)40 b(balls)e(will)f(b)s(e)i(pairwise)f(disjoin)m(t,)i(and)f (the)0 219 y(inner)e(core)f(of)h(eac)m(h)g(them)e(will)g(in)m(tersect)i (with)f(only)g(three)g(spheres)i(from)e(the)g(sequence,)i(namely)0 339 y(the)c(three)f(spheres)j(that)c(in)m(tersect)i(at)f(its)g(cen)m (ter.)120 460 y(F)-8 b(or)28 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Fh(1)3599 5400 y Fu(.)1908 5639 y Fs(17)p eop %%Page: 18 18 18 17 bop 152 100 a Fu(\(3\))49 b(The)34 b(v)m(ectors)f(in)g Fo(O)1064 115 y Fn(i)1131 100 y Fu(are)g(tangen)m(t)g(to)g Fo(\033)1842 63 y Fk(1)1838 126 y Fn(i)1886 100 y Fu(.)152 219 y(\(4\))49 b Fo(O)408 183 y Fk(1)405 244 y(1)474 219 y Fm([)23 b(\001)17 b(\001)g(\001)j([)j Fo(O)869 183 y Fk(1)866 244 y Fn(m)937 254 y Fh(1)1014 219 y Fu(is)33 b Fo("=)p Fu(2)g(dense)i(in)e Fo(G)1759 234 y Fk(0)1832 219 y Fu(:=)28 b Fo(T)2037 183 y Fk(1)2081 219 y Fo(S)2148 183 y Fk(3)2193 219 y Fu(.)152 345 y(\(5\))49 b(The)24 b(pair)f(of)h(an)m(tip)s(o)s(dal)f(p)s(oin)m(ts)h(dual)g(to)f(\006)1945 309 y Fk(1)1945 369 y Fn(m)2016 379 y Fh(1)2083 345 y Fu(is)g(disjoin)m(t)h(from)f(all)g(the)h(spheres)h(\006)3447 309 y Fk(0)3447 370 y(1)3492 345 y Fo(;)17 b(:)g(:)g(:)d(;)j Fu(\006)3786 309 y Fk(1)3786 369 y Fn(m)3857 379 y Fh(1)329 470 y Fu(except)33 b(for)h(\006)866 434 y Fk(1)866 495 y Fn(m)937 505 y Fh(1)975 495 y Fi(\000)p Fk(1)1082 470 y 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b(to)f(whic)m(h)h(the)g(\014rst)g(of)g(these)g(orbits)f(is)h(tangen)m (t)g(mak)m(es)e(angle)i(greater)f(than)h Fo(\031)t(=)p Fu(4)e(with)i(the)0 5400 y(sphere)c(\006)381 5364 y Fk(1)381 5425 y Fn(m)452 5435 y Fh(1)495 5400 y Fu(.)44 b(No)m(w)31 b(w)m(e)g(can)h(rep)s(eatedly)f(apply)g(Prop)s(osition)g(4.5)f(to)h(c)m (ho)s(ose)h(a)f(sequence)i(of)f(crosses)1908 5639 y Fs(18)p eop %%Page: 19 19 19 18 bop 0 100 a Fu(cen)m(tered)35 b(at)e(the)g(great)g(circles)h(to)f (whic)m(h)h(these)g(orbits)f(are)g(tangen)m(t)h(so)f(that:)152 247 y(\(1\))49 b(None)33 b(of)h(spheres)h(in)e(the)g(crosses)i(en)m (ters)f(the)g(cores)g(of)f(the)h(balls)f Fo(B)2978 211 y Fk(1)2973 272 y(1)3022 247 y Fo(;)17 b(:)g(:)g(:)d(;)j(B)3325 211 y Fk(1)3320 272 y Fn(m)3391 282 y Fh(1)3429 272 y Fi(\000)p Fk(1)3536 247 y Fu(.)152 373 y(\(2\))49 b(The)34 b(\014rst)g(cross)h(\(whic)m(h)f(will)f(b)s(e)h(called)g(\(\006)2029 337 y Fk(2)2029 398 y(2)2073 373 y Fo(;)17 b Fu(\006)2190 337 y Fk(2)2190 398 y(3)2234 373 y 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y Fk(2)1660 916 y(1)1705 891 y Fo(;)17 b(:)g(:)g(:)d(;)j Fu(\006)1999 855 y Fk(2)1999 916 y Fn(m)2070 926 y Fh(2)2146 891 y Fu(suc)m(h)35 b(that)1323 1124 y(\006)1395 1083 y Fk(0)1395 1149 y(1)1439 1124 y Fo(;)17 b Fu(\006)1556 1083 y Fk(1)1556 1149 y(1)1600 1124 y Fo(;)g(:)g(:)g(:)e(;)i Fu(\006)1895 1083 y Fk(1)1895 1149 y Fn(m)1966 1159 y Fh(1)2008 1124 y Fo(;)g Fu(\006)2125 1083 y Fk(2)2125 1149 y(1)2169 1124 y Fo(;)g(:)g(:)g(:)d(;)j Fu(\006)2463 1083 y Fk(2)2463 1149 y Fn(m)2534 1159 y Fh(2)0 1346 y Fu(is)49 b(a)f(sequence)i(of)f(orthogonal)f(spheres.)92 b(As)49 b(in)g(the)g(\014rst)g(step,)j(this)d(sequence)h(can)g(b)s(e)f (made)0 1466 y(nondegenerate)f(b)m(y)e(arbitrarily)f(small)f(p)s (erturbations,)50 b(and)d(w)m(e)f(can)h(p)s(erturb)g(\006)3267 1430 y Fk(2)3267 1491 y Fn(m)3338 1501 y Fh(2)3428 1466 y Fu(among)e(the)0 1592 y(great)c(spheres)i(orthogonal)e(to)g(\006)1330 1555 y Fk(2)1330 1616 y Fn(m)1401 1626 y Fh(2)1440 1616 y Fi(\000)p Fk(1)1588 1592 y Fu(so)g(that)g(the)h(dual)g(of)g(\006)2546 1555 y Fk(2)2546 1616 y Fn(m)2617 1626 y Fh(2)2702 1592 y Fu(do)s(es)f(not)h(coincide)g(with)f(an)m(y)0 1717 y(of)36 b(the)h(pairs)f(of)h(an)m(tip)s(o)s(dal)f(p)s(oin)m(ts)g(where) h(three)g(spheres)h(from)e(the)g(sequence)i(in)m(tersect.)54 b(The)37 b Fo(\033)3856 1681 y Fk(2)3852 1743 y Fn(i)0 1837 y Fu(are)d(c)m(hosen)h(to)e(b)s(e)g(the)h(great)f(circles)h(asso)s (ciated)g(with)f(\006)2244 1801 y Fk(1)2244 1862 y Fn(m)2315 1872 y Fh(1)2359 1837 y Fo(;)17 b Fu(\006)2476 1801 y Fk(2)2476 1862 y(1)2520 1837 y Fo(;)g(:)g(:)g(:)d(;)j Fu(\006)2814 1801 y Fk(2)2814 1862 y Fn(m)2885 1872 y Fh(2)2961 1837 y Fu(and)34 b(the)g(orbits)f Fo(O)3689 1801 y Fk(2)3686 1863 y Fn(i)3767 1837 y Fu(are)0 1963 y(c)m(hosen)41 b(so)e(that)g(the)g(v)m(ectors)h(in)f Fo(O)1398 1926 y Fk(2)1395 1989 y Fn(i)1482 1963 y Fu(are)g(tangen)m(t) g(to)g Fo(\033)2211 1926 y Fk(2)2207 1989 y Fn(i)2295 1963 y Fu(and)g(the)h(orbits)f Fo(O)3040 1926 y Fk(2)3037 1989 y Fn(i)3123 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