Content-Type: multipart/mixed; boundary="-------------0303210633719" This is a multi-part message in MIME format. ---------------0303210633719 Content-Type: text/plain; name="03-127.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-127.keywords" reccurence,Sturmian shifts,return time ---------------0303210633719 Content-Type: application/postscript; name="sturm.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="sturm.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86f Copyright 2001 Radical Eye Software %%Title: sturm.dvi %%Pages: 11 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -o sturm.ps sturm.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2003.03.19:1908 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul 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(inequalit)n(y)f(is)g(not)g(satis\014ed)g(for)g(systems)f(with)i(zero)e (en)n(trop)n(y)-7 b(.)53 b(In)456 4234 y(particular)23 b(for)i(the)h(Fib)r(onacci)f(shift)h(obtaind)f(from)f(the)i(golden)e (angle)h(rotation,)f(the)i(lo)n(w)n(er)456 4345 y(lo)r(cal)d(rate)h (assumes)f(the)i(v)-5 b(alue)24 b Fx(r)1522 4357 y Fm(0)1583 4345 y Ft(=)1680 4312 y Fm(3)p Fn(\000)1765 4264 y(p)p 1820 4264 34 3 v 48 x Fm(5)p 1680 4326 173 4 v 1750 4374 a(2)1886 4345 y Fr(<)e Ft(1.)36 b(Afraimo)n(vic)n(h)23 b(et)h(al)g([)p Fx(?)p Ft(])h(sho)n(w)e(that)i Fx(r)3255 4357 y Fm(0)3315 4345 y Ft(=)e(0)456 4445 y(for)h(some)g(rotations)g (of)h(the)g(circle)g(whose)f(parameter)g(has)g(un)n(b)r(ounded)i(con)n (tin)n(ued)e(fraction)456 4545 y(expansion.)56 b(It)35 b(follo)n(ws)e(that)i(the)f(same)g(result)g(holds)g(for)g(the)h (corresp)r(onding)d(Sturmian)456 4644 y(subshift.)43 b(K)-8 b(\027)-54 b(urk)-5 b(a)29 b([)p Fx(?)p Ft(])h(treats)f(the)h (case)e(of)i(substitutiv)n(e)g(subshifts)g(and)f(obtains)g(a)g(form)n (ula)456 4744 y(for)e Fx(r)622 4756 y Fm(0)687 4744 y Ft(and)g Fx(r)887 4756 y Fm(1)925 4744 y Ft(.)37 b(In)28 b(this)g(case)e(b)r(oth)i Fx(r)1661 4756 y Fm(0)1727 4744 y Ft(and)f Fx(r)1927 4756 y Fm(1)1992 4744 y Ft(are)g(p)r(ositiv)n (e)g(and)g(\014nite.)p 456 4875 499 4 v 555 4969 a Fl(Date)5 b Fw(:)23 b(Marc)n(h)h(19,)f(2003.)555 5050 y(F)-6 b(acult)n(y)25 b(of)e(Mathematics)h(and)g(Ph)n(ysics,)g(Charles)f(Univ)n(ersit)n(y)h (in)f(Prague)456 5133 y(Malostransk)n(\023)-33 b(e)24 b(n\023)-35 b(am)n(\024)i(est)-8 b(\023)-27 b(\020)23 b(25,)h(CZ-11800)h(Praha)f(1,)g(Czec)n(hia)g(and)h(Cen)n(tre)g(de)f(Ph) n(ysique)g(Th)n(\023)-33 b(eorique,)24 b(CNRS)456 5216 y(Lumin)n(y)-6 b(,)22 b(F-13288)j(Marseille)d(cedex)j(9.)1933 5315 y Fm(1)p eop %%Page: 2 2 2 1 bop 456 251 a Fm(2)1200 b(MICHAL)22 b(KUPSA)555 450 y Ft(In)37 b(this)g(pap)r(er)f(w)n(e)g(will)g(discuss)g(completely)h (the)g(situation)f(in)h(the)g(Sturmian)f(shifts.)456 550 y(One)26 b(can)h(easy)f(c)n(hec)n(k)g(that)h(the)h(result)e(of)h (Afraimo)n(vic)n(h)f(et)h(al)g(considered)f(for)g(corresp)r(ond-)456 649 y(ing)37 b(Sturmian)g(shifts)g(and)g(the)h(result)f(of)g(Cassaigne) e(et)i(al)g(for)g(Fib)r(onacci)g(shift)g(follo)n(ws)456 749 y(immediately)-7 b(.)57 b(W)-7 b(e)35 b(giv)n(e)e(form)n(ulas)g (for)h Fx(r)1818 761 y Fm(0)1890 749 y Ft(and)g Fx(r)2097 761 y Fm(1)2169 749 y Ft(in)h(terms)f(of)g(the)h(con)n(v)n(ergen)n(ts)d Fr(q)3254 761 y Fq(k)3329 749 y Ft(ob-)456 849 y(tained)f(from)g(the)h (con)n(tin)n(ued)f(fraction)f(expansion)h(of)g(the)h(parameter)d Fr(\013)h Ft(=)f([0)p Fr(;)14 b(a)3110 861 y Fm(1)3146 849 y Fr(;)g(a)3227 861 y Fm(2)3265 849 y Fr(;)g(:)g(:)g(:)o Ft(].)456 948 y(If)23 b Fr(a)578 960 y Fq(k)642 948 y Ft(are)f(b)r(ounded,)j(then)f Fx(r)1359 960 y Fm(0)1420 948 y Ft(and)f Fx(r)1616 960 y Fm(1)1677 948 y Ft(are)f(p)r(ositiv)n(e) h(and)g(\014nite.)36 b(If)23 b Fr(a)2636 960 y Fq(k)2700 948 y Ft(are)f(un)n(b)r(ounded,)j(then)456 1048 y Fx(r)495 1060 y Fm(0)555 1048 y Ft(=)e(0)k(and)g Fx(r)912 1060 y Fm(1)973 1048 y Ft(=)c Fp(1)p Ft(.)1564 1227 y(2.)41 b Fs(Sturmian)31 b(shifts)555 1376 y Ft(A)c(dynamical)e(system)h(is)g (a)g(pair)g(\()p Fr(X)r(;)14 b(F)e Ft(\),)26 b(where)g Fr(X)32 b Ft(is)26 b(a)g(compact)g(metric)g(space)f(and)h Fr(F)456 1476 y Ft(is)31 b(a)h(con)n(tin)n(uous)e(function)j(from)e Fr(X)38 b Ft(to)32 b Fr(X)7 b Ft(.)49 b(The)31 b(P)n(oincar)n(\023)-39 b(e)29 b(recurrence)h(time)i(of)g(a)f(subset)456 1575 y Fr(M)g Fp(\022)23 b Fr(X)34 b Ft(is)1242 1678 y Fr(\034)9 b Ft(\()p Fr(M)g Ft(\))24 b(=)e(min)q Fp(f)p Fr(k)j(>)e Ft(0)f Fp(j)i Fr(F)2065 1644 y Fq(k)2105 1678 y Ft(\()p Fr(M)9 b Ft(\))19 b Fp(\\)g Fr(M)31 b Fp(6)p Ft(=)23 b Fp(;g)p Fr(:)456 1798 y Ft(Let)h Fr(A)g Ft(b)r(e)g(a)g(\014nite)g (alphab)r(et,)h(and)e Fr(A)1653 1768 y Fo(N)1724 1798 y Ft(the)h(space)f(of)h(all)g(in\014nite)g(sequences)f(of)h(letters)g (from)456 1898 y Fr(A)33 b Ft(with)g(the)h(pro)r(duct)f(top)r(ology)-7 b(.)51 b(The)33 b(set)g Fr(A)1966 1868 y Fn(\003)2038 1898 y Ft(consists)f(of)h(all)g(w)n(ords)e(\(\014nite)j(sequences\))456 1998 y(o)n(v)n(er)27 b Fr(A)p Ft(.)42 b(F)-7 b(or)29 b(a)g(w)n(ord)f Fr(u)d Ft(=)h Fr(u)1403 2010 y Fm(0)1439 1998 y Fr(u)1487 2010 y Fm(2)1538 1998 y Fr(:)14 b(:)g(:)g(u)1697 2010 y Fq(n)p Fn(\000)p Fm(1)1852 1998 y Fp(2)26 b Fr(A)1995 1968 y Fn(\003)2034 1998 y Ft(,)k(denote)f(b)n(y)g Fp(j)p Fr(u)p Fp(j)c Ft(=)h Fr(n)j Ft(its)g(length.)42 b(The)29 b(set)456 2097 y Fr(A)518 2067 y Fq(n)595 2097 y Ft(consists)i(of)h (all)g(w)n(ords)e(of)i(length)g Fr(n)p Ft(.)50 b(The)32 b(shift)h(map)e Fr(\033)j Ft(:)d Fr(A)2598 2067 y Fo(N)2674 2097 y Fp(\000)-14 b(!)30 b Fr(A)2900 2067 y Fo(N)2979 2097 y Ft(is)h(de\014ned)i(b)n(y)456 2197 y Fr(\033)503 2209 y Fq(i)531 2197 y Ft(\()p Fr(x)p Ft(\))24 b(=)e Fr(x)800 2209 y Fq(i)p Fm(+1)913 2197 y Ft(.)555 2297 y(A)33 b(shift)g(is)f(an)n(y)g(subsystem)g(\(\006)p Fr(;)14 b(\033)s Ft(\))33 b(of)f(\()p Fr(A)1934 2266 y Fo(N)1981 2297 y Fr(;)14 b(\033)s Ft(\),)34 b(where)e(\006)f Fp(\022)g Fr(A)2651 2266 y Fo(N)2729 2297 y Ft(is)i(nonempt)n(y)-7 b(,)33 b(closed)456 2396 y(and)e Fr(\033)s Ft(-in)n(v)-5 b(arian)n(t.)49 b(F)-7 b(or)32 b(a)f(shift)i(\006)f(and)g(for)f(a)h(w)n (ord)f Fr(u)f Ft(=)g Fr(u)2406 2408 y Fm(0)2442 2396 y Fr(u)2490 2408 y Fm(1)2541 2396 y Fr(:)14 b(:)g(:)g(u)2700 2408 y Fq(n)p Fn(\000)p Fm(1)2860 2396 y Fp(2)30 b Fr(A)3007 2366 y Fn(\003)3078 2396 y Ft(w)n(e)i(denote)456 2496 y(b)n(y)f([)p Fr(u)p Ft(])e(=)g Fp(f)p Fr(x)h Fp(2)g Ft(\006)g Fp(j)f(8)p Fr(i)g(<)g(n)g Ft(:)h Fr(x)1516 2508 y Fq(i)1574 2496 y Ft(=)f Fr(u)1716 2508 y Fq(i)1743 2496 y Fp(g)i Ft(the)h(cylinder)f(of)h Fr(u)p Ft(.)48 b(The)32 b(language)e(of)h(a)g(shift)i(is)456 2595 y(the)d(set)h(of)f (w)n(ords)f(whic)n(h)h(ha)n(v)n(e)f(nonempt)n(y)h(cylinders,)h(i.e.,)45 b Fp(L)p Ft(\(\006\))28 b(=)f Fp(f)p Fr(u)g Fp(2)g Fr(A)3008 2565 y Fn(\003)3074 2595 y Fp(j)h Ft([)p Fr(u)p Ft(])f Fp(6)p Ft(=)g Fp(;g)p Ft(.)456 2695 y(The)k(set)45 b Fp(L)834 2659 y Fq(n)879 2695 y Ft(\(\006\))32 b(consists)e(of)i(all)e (w)n(ords)g(of)h(the)h(language)d(of)j(length)f Fr(n)p Ft(.)47 b(If)32 b(w)n(e)f(denote)g(b)n(y)456 2795 y Fr(x)p Ft(\()p Fr(n)p Ft(\))24 b(=)e Fr(x)775 2807 y Fm(0)813 2795 y Fr(x)860 2807 y Fm(1)911 2795 y Fr(:)14 b(:)g(:)g(x)1069 2807 y Fq(n)p Fn(\000)p Fm(1)1224 2795 y Ft(the)26 b(pre\014x)e(of)g Fr(x)g Fp(2)f Ft(\006)i(of)g(length)f Fr(n)p Ft(,)i(then)39 b Fp(L)2618 2758 y Fq(n)2663 2795 y Ft(\(\006\))24 b(=)e Fp(f)p Fr(x)p Ft(\()p Fr(n)p Ft(\))i Fp(j)f Fr(x)g Fp(2)h Ft(\006)p Fp(g)p Ft(.)555 2894 y(A)i(Sturmian)f(shift)h(is)f(a)g(co)r (ding)g(of)g(an)g(irrational)f(rotation)g(of)h(the)h(unit)g(circle)e (\(Hedlund)456 2994 y(and)30 b(Morse)g([)p Fx(?)p Ft(]\).)46 b(This)31 b(is)f(a)g(dynamical)g(system)h(\()p Fk(T)p Fr(;)14 b(F)2273 3006 y Fq(\013)2320 2994 y Ft(\),)32 b(where)e Fk(T)d Ft(=)h([0)p Fr(;)14 b Ft(1[)30 b(is)g(the)h(circle)456 3094 y(with)k(the)f(metric)h Fr(d)p Ft(\()p Fr(x;)14 b(y)s Ft(\))35 b(=)f(min)q Fp(fj)p Fr(x)23 b Fp(\000)f Fr(y)s Fp(j)p Fr(;)14 b Ft(1)22 b Fp(\000)h(j)p Fr(x)h Fp(\000)e Fr(y)s Fp(jg)34 b Ft(and)g Fr(F)2598 3106 y Fq(\013)2646 3094 y Ft(\()p Fr(x)p Ft(\))i(=)e Fr(x)23 b Ft(+)g Fr(\013)55 b Ft(mo)r(d)63 b(1,)456 3193 y(where)27 b Fr(\013)c Fp(2)h Fk(R)p Ft(.)42 b(W)-7 b(e)28 b(consider)f(only)g (irrational)f(angles)h(from)g(the)h(op)r(en)f(in)n(terv)-5 b(al)27 b Fr(\013)d Fp(2)p Ft(]0)p Fr(;)14 b Ft(1[.)555 3293 y(There)35 b(is)f(the)i(canonical)d(partition)49 b Fp(I)41 b Ft(=)35 b Fp(f)p Fr(I)2047 3305 y Fm(0)2084 3293 y Fr(;)14 b(I)2157 3305 y Fm(1)2195 3293 y Fp(g)35 b Ft(of)f Fk(T)p Ft(,)j(where)d Fr(I)2772 3305 y Fm(0)2845 3293 y Ft(=)h([0)p Fr(;)14 b Ft(1)22 b Fp(\000)h Fr(\013)p Ft([)36 b(and)456 3392 y Fr(I)492 3404 y Fm(1)552 3392 y Ft(=)23 b([1)18 b Fp(\000)g Fr(\013;)c Ft(1[.)37 b(F)-7 b(or)27 b Fr(u)22 b Fp(2)i Fx(2)1367 3362 y Fn(\003)1405 3392 y Ft(,)k(set)1585 3622 y Fr(I)1621 3634 y Fq(u)1688 3622 y Ft(=)1776 3512 y Fn(j)p Fq(u)p Fn(j\000)p Fm(1)1812 3544 y Fj(\\)1797 3722 y Fq(k)q Fm(=0)1953 3622 y Fr(F)2018 3588 y Fn(\000)p Fq(k)2006 3643 y(\013)2111 3622 y Ft(\()p Fr(I)2179 3634 y Fq(u)2218 3643 y Fi(k)2259 3622 y Ft(\))p Fr(:)456 3835 y Ft(An)n(y)33 b Fr(I)675 3847 y Fq(u)753 3835 y Ft(is)g(either)h(a)f(semiop)r(en)g(in)n(terv)-5 b(al)33 b(or)g(the)h(empt)n(y)g(set.)55 b(The)33 b(asso)r(ciated)f (Sturmian)456 3934 y(shift)26 b(\(\006)733 3946 y Fq(\013)780 3934 y Fr(;)14 b(\033)s Ft(\))27 b(is)e(de\014ned)h(b)n(y)f(its)h (language)38 b Fp(L)p Ft(\(\006)2025 3946 y Fq(\013)2073 3934 y Ft(\))23 b(=)g Fp(f)p Fr(u)f Fp(2)h Fx(2)2454 3904 y Fn(\003)2515 3934 y Fp(j)g Fr(I)2597 3946 y Fq(u)2664 3934 y Fp(6)p Ft(=)g Fp(;g)p Ft(.)35 b(In)26 b(other)f(w)n(ords,)1323 4076 y(\006)1383 4088 y Fq(\013)1453 4076 y Ft(=)e Fp(f)p Fr(x)g Fp(2)h Fx(2)1780 4042 y Fo(N)1848 4076 y Fp(j)g(8)p Fr(n)e Fp(2)h Fk(N)t Fr(;)14 b(I)2219 4091 y Fq(x)p Fm(\()p Fq(n)p Fm(\))2383 4076 y Fp(6)p Ft(=)23 b Fp(;g)p Fr(:)456 4214 y Ft(If)30 b Fr(\013)c Fp(2)p 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Ft(\()p Fr(x)p Ft(\))84 b(=)f(lim)14 b(inf)1885 5009 y Fq(n)p Fn(!1)2111 4897 y Fr(\034)9 b Ft(\()p Fr(I)2224 4912 y Fq(x)p Fm(\()p Fq(n)p Fm(\))2360 4897 y Ft(\))p 2111 4940 282 4 v 2227 5016 a Fr(n)p 1451 5108 64 4 v 1451 5175 a(R)p Ft(\()p Fr(x)p Ft(\))84 b(=)f(lim)14 b(sup)1897 5240 y Fq(n)p Fn(!1)2135 5113 y Fr(\034)9 b Ft(\()p Fr(I)2248 5128 y Fq(x)p Fm(\()p Fq(n)p Fm(\))2384 5113 y Ft(\))p 2135 5155 282 4 v 2251 5231 a Fr(n)2426 5175 y(:)p eop %%Page: 3 3 3 2 bop 1173 251 a Fm(LOCAL)29 b(RETURN)h(RA)-5 b(TES)28 b(IN)i(STURMIAN)f(SHIFTS)683 b(3)555 450 y Ft(The)33 b(description)f(of)g(the)h(in)n(terv)-5 b(als)31 b Fr(I)1781 462 y Fq(u)1858 450 y Ft(is)h(obtained)g(from)g(the)h(con)n(tin)n(ued)f (fraction)g(ex-)456 550 y(pansion)k(of)h Fr(\013)p Ft(.)66 b(There)37 b(exists)g(a)g(unique)g(sequence)g Fp(f)p Fr(a)2303 562 y Fq(k)2343 550 y Fp(g)2385 520 y Fn(1)2385 573 y Fq(k)q Fm(=1)2547 550 y Ft(of)g(p)r(ositiv)n(e)g(in)n(tegers)f (suc)n(h)456 649 y(that)1250 791 y Fr(\013)23 b Ft(=)g([0)p Fr(;)14 b(a)1560 803 y Fm(1)1597 791 y Fr(;)g(a)1678 803 y Fm(2)1715 791 y Fr(;)g(:)g(:)g(:)o Ft(])24 b(=)e(0)c(+)2355 734 y(1)p 2135 772 482 4 v 2135 891 a Fr(a)2179 903 y Fm(1)2235 891 y Ft(+)2447 835 y(1)p 2328 872 280 4 v 2328 948 a Fr(a)2372 960 y Fm(2)2427 948 y Ft(+)g Fr(:)c(:)g(:)2627 791 y(:)456 1072 y Ft(The)32 b(con)n(v)n(ergen)n(ts)e(of)i Fr(\013)h Ft(are)f(the)g(sequences)g Fp(f)p Fr(p)2026 1084 y Fq(k)2066 1072 y Fp(g)2108 1042 y Fn(1)2108 1095 y Fq(k)q Fm(=)p Fn(\000)p Fm(1)2284 1072 y Ft(,)i Fp(f)p Fr(q)2420 1084 y Fq(k)2461 1072 y Fp(g)2503 1042 y Fn(1)2503 1095 y Fq(k)q Fm(=)p Fn(\000)p Fm(1)2711 1072 y Ft(de\014ned)f(b)n(y)f Fr(p)3164 1084 y Fn(\000)p Fm(1)3284 1072 y Ft(=)f(1,)456 1172 y Fr(q)493 1184 y Fn(\000)p Fm(1)605 1172 y Ft(=)22 b(0,)28 b Fr(p)827 1184 y Fm(0)887 1172 y Ft(=)22 b(0,)28 b Fr(q)1104 1184 y Fm(0)1164 1172 y Ft(=)23 b(1)k(and)1088 1329 y Fr(q)1125 1341 y Fq(k)q Fm(+1)1273 1329 y Ft(=)22 b Fr(a)1404 1341 y Fq(k)q Fm(+1)1529 1329 y Fr(q)1566 1341 y Fq(k)1626 1329 y Ft(+)c Fr(q)1746 1341 y Fq(k)q Fn(\000)p Fm(1)1871 1329 y Fr(;)97 b(p)2033 1341 y Fq(k)q Fm(+1)2181 1329 y Ft(=)23 b Fr(a)2313 1341 y Fq(k)q Fm(+1)2438 1329 y Fr(p)2480 1341 y Fq(k)2539 1329 y Ft(+)18 b Fr(p)2664 1341 y Fq(k)q Fn(\000)p Fm(1)2789 1329 y Fr(:)456 1487 y Ft(By)g(the)i(Klein)f(theorem)f(\(see)h(Hardy)f(and)h(W)-7 b(righ)n(t)19 b(HW\),)h(the)f(closest)g(returns)f(of)h(the)g(iterates) 456 1587 y Fr(F)521 1557 y Fq(n)509 1607 y(\013)566 1587 y Ft(\(0\))30 b(to)h(zero)e(happ)r(en)i(at)g(times)f Fr(q)1647 1599 y Fq(k)1688 1587 y Ft(.)46 b(W)-7 b(e)31 b(ha)n(v)n(e)f Fr(d)p Ft(\(0)p Fr(;)14 b(F)2317 1557 y Fq(q)2347 1566 y Fi(k)2387 1587 y Ft(\(0\)\))29 b(=)e Fr(\021)2687 1599 y Fq(k)2756 1587 y Ft(=)h(\()p Fp(\000)p Ft(1\))3020 1557 y Fq(k)3060 1587 y Ft(\()p Fr(q)3129 1599 y Fq(k)3171 1587 y Fr(\013)20 b Fp(\000)g Fr(p)3371 1599 y Fq(k)3412 1587 y Ft(\))456 1686 y(and)27 b(for)g Fr(q)781 1698 y Fq(k)845 1686 y Fr(<)c(n)g(<)f(q)1130 1698 y Fq(k)q Fm(+1)1255 1686 y Ft(,)28 b Fr(d)p Ft(\(0)p Fr(;)14 b(F)1525 1656 y Fq(n)1570 1686 y Ft(\(0\)\))23 b Fr(>)g(\021)1860 1698 y Fq(k)1901 1686 y Ft(.)37 b(In)28 b(particular)e Fr(\021)2489 1698 y Fn(\000)p Fm(1)2602 1686 y Ft(=)c(1,)27 b Fr(\021)2822 1698 y Fm(0)2883 1686 y Ft(=)c Fr(\013)28 b Ft(and)1540 1844 y Fr(\021)1581 1856 y Fq(k)q Fm(+1)1730 1844 y Ft(=)22 b Fr(a)1861 1856 y Fq(k)q Fm(+1)1986 1844 y Fr(\021)2027 1856 y Fq(k)2087 1844 y Fp(\000)c Fr(\021)2211 1856 y Fq(k)q Fn(\000)p Fm(1)2337 1844 y Fr(:)456 2002 y Ft(The)36 b(sequence)f Fp(f)p Fr(\021)1070 2014 y Fq(k)1111 2002 y Fp(g)1153 1972 y Fn(1)1153 2026 y Fq(k)q Fm(=)p Fn(\000)p Fm(1)1365 2002 y Ft(is)h(p)r(ositiv)n(e,)i(decreasing)c(and)i(con)n(v)n(erges)d (to)j(zero.)61 b(It)36 b(follo)n(ws)456 2103 y(that)27 b(if)i Fr(I)h Ft(=)22 b([)p Fr(a;)14 b(b)p Ft([)28 b(is)f(a)g(semiop)r (en)h(in)n(terv)-5 b(al,)27 b(then)1359 2261 y Fr(\021)1400 2273 y Fq(k)q Fm(+1)1548 2261 y Fr(<)22 b Fp(j)p Fr(I)7 b Fp(j)24 b(\024)e 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Fj(\032)922 5085 y Ft(\()p Fp(\000)p Ft(1\))1093 5055 y Fq(k)q Fn(\000)p Fm(1)1218 5085 y Ft([)p Fp(h)p Ft(\()p Fr(u)1353 5097 y Fq(k)1413 5085 y Ft(+)18 b(1\))p Fr(q)1607 5097 y Fq(k)q Fn(\000)p Fm(1)1751 5085 y Ft(+)g Fr(a)p Fp(i)p Fr(;)c Fp(h)p Fr(u)2027 5097 y Fq(k)2068 5085 y Fr(q)2105 5097 y Fq(k)q Fn(\000)p Fm(1)2249 5085 y Ft(+)k Fr(a)p Fp(i)p Ft([)85 b Fh(if)h Ft(0)22 b Fp(\024)h Fr(u)2852 5097 y Fq(k)2915 5085 y Fp(\024)g Fr(a)3047 5097 y Fq(k)3106 5085 y Fp(\000)18 b Ft(1)922 5186 y(\()p Fp(\000)p Ft(1\))1093 5156 y Fq(k)q Fn(\000)p Fm(1)1218 5186 y Ft([)p Fp(h)p Fr(b)p Fp(i)p Fr(;)c Fp(h)p Fr(u)1458 5198 y Fq(k)1499 5186 y Fr(q)1536 5198 y Fq(k)q Fn(\000)p Fm(1)1680 5186 y Ft(+)k Fr(a)p Fp(i)p Ft([)654 b Fh(if)86 b Fr(u)2700 5198 y Fq(k)2763 5186 y Ft(=)23 b Fr(a)2895 5198 y Fq(k)p eop %%Page: 5 5 5 4 bop 1173 251 a Fm(LOCAL)29 b(RETURN)h(RA)-5 b(TES)28 b(IN)i(STURMIAN)f(SHIFTS)683 b(5)p 504 1092 2892 4 v 531 1104 4 29 v 900 1104 V 1268 1104 V 1637 1104 V 2133 1104 V 2501 1104 V 2870 1104 V 3366 1104 V 589 1161 a Fr(I)625 1173 y Fm(001010)958 1161 y Fr(I)994 1173 y Fm(010010)1326 1161 y Fr(I)1362 1173 y Fm(010100)1752 1161 y Fr(I)1788 1173 y Fm(010101)2205 1161 y Fr(I)2241 1173 y Fm(100101)2574 1161 y Fr(I)2610 1173 y Fm(101001)2999 1161 y Fr(I)3035 1173 y Fm(101010)476 1217 y Fp(h)p Ft(0)p Fp(i)262 b(h)p Ft(2)p Fp(i)h(h)p Ft(4)p Fp(i)f(h)p Ft(6)p Fp(i)391 b(h)p Ft(1)p Fp(i)262 b(h)p Ft(3)p Fp(i)h(h)p Ft(5)p Fp(i)674 1047 y Fr(J)720 1059 y Fm(20)1043 1047 y Fr(J)1089 1059 y Fm(21)1411 1047 y Fr(J)1457 1059 y Fm(22)1865 1047 y Fr(J)1911 1059 y Fm(23)2290 1047 y Fr(J)2336 1059 y Fm(11)2659 1047 y Fr(J)2705 1059 y Fm(12)3027 1047 y Fr(J)3073 1059 y Fm(13)p 504 808 2892 4 v 531 821 4 29 v 2133 821 V 3366 821 V 1383 877 a Fr(I)1419 889 y Fm(0)2800 877 y Fr(I)2836 889 y Fm(1)1383 764 y Fr(J)1429 776 y Fm(2)2800 764 y Fr(J)2846 776 y Fm(1)476 906 y Fp(h)p Ft(0)p Fp(i)1496 b(h)p Ft(1)p Fp(i)1314 1417 y Fs(Figure)32 b(2.)41 b Ft(P)n(artitions)26 b(of)i(the)g(circle) 555 1627 y(Here)g(\()p Fp(\000)p Ft(1\)[)p Fr(b;)14 b(a)p Ft([=)22 b([)p Fr(a;)14 b(b)p Ft([,)28 b(where)f(0)d Fp(\024)f Fr(a)g(<)h(b)f(<)g Ft(1,)28 b(is)g(a)f(semiop)r(en)h(in)n (terv)-5 b(al)28 b(of)g(the)g(circle.)456 1727 y(W)-7 b(e)26 b(iden)n(tify)h(also)e([)p Fr(a;)14 b Ft(0[=)22 b([)p Fr(a;)14 b Ft(1[=)22 b(\()p Fp(\000)p Ft(1\)[0)p Fr(;)14 b(a)p Ft([.)35 b(If)27 b Fr(x)c Fp(2)h Fr(X)2273 1739 y Fq(\013)2320 1727 y Ft(,)i(w)n(e)g(denote)g(b)n(y)g Fr(x)p Ft(\()p Fr(n)p Ft(\))e(=)f Fr(x)3190 1739 y Fm(1)3241 1727 y Fr(:)14 b(:)g(:)g(x)3399 1739 y Fq(n)456 1826 y Ft(the)28 b(pre\014x)g(of)h Fr(x)g Ft(of)f(length)h Fr(n)p Ft(.)39 b(In)29 b(Figure)e Fx(??)i Ft(w)n(e)f(can)g(see)g(the)h (partitions)f(of)g(the)h(circle)f(for)456 1926 y Fr(\013)23 b Ft(=)g([0)p Fr(;)14 b Ft(2)p Fr(;)g Ft(3)p Fr(;)g(:)g(:)g(:)n Ft(].)456 2082 y Fx(Prop)s(osition)30 b(2.)41 b Fh(If)30 b Fr(u)22 b Fp(2)38 b(L)1395 2046 y Fq(k)1436 2082 y Ft(\()p Fr(X)1537 2094 y Fq(\013)1584 2082 y Ft(\))p Fh(,)31 b Fr(k)25 b Fp(\025)e Ft(1)p Fh(,)30 b(then)1322 2268 y Fp(j)p Fr(J)1391 2280 y Fq(u)1434 2268 y Fp(j)23 b Ft(=)1568 2151 y Fj(\032)1672 2217 y Fr(\021)1713 2229 y Fq(k)q Fn(\000)p Fm(1)2107 2217 y Fh(if)86 b Fr(u)2291 2229 y Fq(k)2354 2217 y Fr(<)23 b(a)2486 2229 y Fq(k)1672 2317 y Fr(\021)1713 2329 y Fq(k)q Fn(\000)p Fm(1)1857 2317 y Ft(+)18 b Fr(\021)1981 2329 y Fq(k)2107 2317 y Fh(if)86 b Fr(u)2291 2329 y Fq(k)2354 2317 y Ft(=)23 b Fr(a)2486 2329 y Fq(k)456 2449 y Fh(and)748 2585 y Fp(I)799 2549 y Fq(q)829 2558 y Fi(k)866 2549 y Fn(\000)p Fm(1)1038 2585 y Ft(=)83 b Fp(f)p Fr(J)1274 2597 y Fq(u)1340 2585 y Fp(j)23 b Fr(u)f Fp(2)38 b(L)1606 2549 y Fq(k)1647 2585 y Ft(\()p Fr(X)1748 2597 y Fq(\013)1795 2585 y Ft(\))p Fp(g)1038 2713 y Ft(=)83 b Fp(f)p Ft(\()p Fp(\000)p Ft(1\))1399 2679 y Fq(k)q Fn(\000)p Fm(1)1524 2713 y Ft([)p Fp(h)p Fr(i)18 b Ft(+)g Fr(q)1746 2725 y Fq(k)q Fn(\000)p Fm(1)1872 2713 y Fp(i)p Fr(;)c Fp(h)p Fr(i)p Fp(i)p Ft([)p Fp(j)23 b Fr(i)g Ft(=)g(0)p 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Fp(j)g Fr(j)34 b(<)28 b(a)2957 3822 y Fq(k)q Fm(+1)3081 3810 y Fp(g)j Ft(coincide)456 3910 y(and)37 b(w)n(e)g(ha)n(v)n(e)f(pro) n(v)n(ed)g(that)h(its)h(length)f(is)h Fr(\021)1953 3922 y Fq(k)1994 3910 y Ft(.)66 b(Denote)37 b Fr(J)48 b Ft(=)2575 3847 y Fj(S)2658 3910 y Fr(M)9 b Ft(.)65 b(If)38 b Fr(u)2977 3922 y Fq(k)3057 3910 y Fr(<)h(a)3205 3922 y Fq(k)3283 3910 y Ft(then)456 4009 y Fp(j)p Fr(J)8 b Fp(j)31 b Ft(=)g(\()p Fr(a)759 4021 y Fq(k)q Fm(+1)906 4009 y Fp(\000)22 b Ft(1\))p Fr(\021)1108 4021 y Fq(k)1181 4009 y Ft(and)33 b(if)g Fr(u)1477 4021 y Fq(k)1549 4009 y Ft(=)e Fr(a)1689 4021 y Fq(k)1762 4009 y Ft(then)i Fp(j)p Fr(J)8 b Fp(j)32 b Ft(=)f Fr(a)2228 4021 y Fq(k)q Fm(+1)2353 4009 y Fr(\021)2394 4021 y Fq(k)2435 4009 y Ft(.)52 b(In)33 b(b)r(oth)g(cases,)g Fp(j)p Fr(J)8 b Fp(j)32 b Fr(<)f Fp(j)p Fr(J)3355 4021 y Fq(u)3398 4009 y Fp(j)p Ft(,)456 4109 y Fr(J)502 4121 y Fq(ua)577 4130 y Fi(k)q Fg(+1)717 4109 y Ft(=)e Fr(J)857 4121 y Fq(u)921 4109 y Fp(\000)20 b Fr(J)40 b Ft(and)31 b Fp(j)p Fr(J)1326 4121 y Fq(ua)1401 4130 y Fi(k)q Fg(+1)1512 4109 y Fp(j)e Ft(=)g Fr(\021)1699 4121 y Fq(k)q Fn(\000)p Fm(1)1846 4109 y Fp(\000)20 b Ft(\()p Fr(a)2007 4121 y Fq(k)q Fm(+1)2153 4109 y Fp(\000)h Ft(1\))p Fr(\021)2354 4121 y Fq(k)2423 4109 y Ft(=)29 b Fr(\021)2558 4121 y Fq(k)2620 4109 y Ft(+)20 b Fr(\021)2746 4121 y Fq(k)q Fm(+1)2871 4109 y Ft(.)48 b(Th)n(us)d Fp(J)3241 4121 y Fq(u)3316 4109 y Ft(is)31 b(a)456 4222 y(partition)c(of)g Fr(J)942 4234 y Fq(u)986 4222 y Ft(.)37 b(Because)26 b Fp(f)p Fr(J)1451 4234 y Fq(u)1517 4222 y Fp(j)d Fr(u)g Fp(2)37 b(L)1784 4186 y Fq(k)1824 4222 y Ft(\()p Fr(X)1925 4234 y Fq(\013)1973 4222 y Ft(\))p Fp(g)27 b Ft(is)h(a)f(partition)g (of)h Fk(T)p Ft(,)f(then)h(also)1197 4374 y Fp(J)38 b Ft(=)23 b Fp(f)p Fr(J)1467 4386 y Fq(v)1529 4374 y Fp(j)g Fr(v)j Fp(2)38 b(L)1791 4338 y Fq(k)q Fm(+1)1916 4374 y Ft(\()p Fr(X)2017 4386 y Fq(\013)2064 4374 y Ft(\))p Fp(g)23 b Ft(=)2365 4296 y Fj([)2249 4482 y Fq(u)p Fn(2)11 b(L)2390 4463 y Fi(k)2426 4482 y Fm(\()p Fq(X)2506 4490 y Fi(\013)2549 4482 y Fm(\))2602 4374 y Fp(J)2674 4386 y Fq(u)456 4607 y Ft(is.)62 b(It)36 b(is)g(not)h(di\016cult)f(to)g(pro) n(v)n(e)f(that)h(the)h(endp)r(oin)n(ts)f(of)g(in)n(terv)-5 b(als)35 b(from)50 b Fp(J)i Ft(b)r(elong)36 b(to)456 4707 y Fp(fh)p Fr(i)p Fp(i)29 b(j)h Ft(0)g Fp(\024)g Fr(i)f Fp(\024)h Fr(q)1030 4719 y Fq(k)q Fm(+1)1176 4707 y Fp(\000)21 b Ft(1)p Fp(g)p Ft(.)49 b(The)32 b(partitions)44 b Fp(J)k Ft(and)d Fp(I)2324 4670 y Fq(q)2354 4679 y Fi(k)q Fg(+1)2461 4670 y Fn(\000)p Fm(1)2582 4707 y Ft(con)n(tain)32 b(in)n(terv)-5 b(als)30 b(of)i(t)n(w)n(o)456 4809 y(lengths)27 b Fr(\021)783 4821 y Fq(k)852 4809 y Ft(and)g Fr(\021)1054 4821 y Fq(k)1114 4809 y Ft(+)18 b Fr(\021)1238 4821 y Fq(k)q Fm(+1)1363 4809 y Ft(,)28 b(hence)41 b Fp(J)e Ft(=)d Fp(I)1906 4773 y Fq(q)1936 4782 y Fi(k)q Fg(+1)2043 4773 y Fn(\000)p Fm(1)2132 4809 y Ft(.)h(F)-7 b(or)27 b(the)h(partition)889 4948 y Fp(J)961 4912 y Fn(0)1067 4948 y Ft(=)83 b Fp(f)p Ft(\()p Fp(\000)p Ft(1\))1428 4914 y Fq(k)1468 4948 y Ft([)p Fp(h)p Fr(i)18 b Ft(+)g Fr(q)1690 4960 y Fq(k)1731 4948 y Fp(i)p Fr(;)c Fp(h)p Fr(i)p Fp(i)p Ft([)p Fp(j)24 b Fr(i)e Ft(=)h(0)p Fr(;)14 b Ft(1)p Fr(;)g(:)g(:)g(:)f(;)h(q)2444 4960 y Fq(k)q Fm(+1)2587 4948 y Fp(\000)k Fr(q)2707 4960 y Fq(k)2766 4948 y Fp(\000)g Ft(1)p Fp(g)g([)1215 5077 y(f)p Ft(\()p Fp(\000)p Ft(1\))1428 5042 y Fq(k)1468 5077 y Ft([)p Fp(h)p Fr(i)p Fp(i)p Fr(;)c Fp(h)p Fr(i)k Ft(+)g Fr(q)1820 5089 y Fq(k)q Fm(+1)1964 5077 y Fp(\000)g Fr(q)2084 5089 y Fq(k)2125 5077 y Fp(i)p Ft([)p Fp(j)23 b Fr(i)g Ft(=)f(0)p Fr(;)14 b Ft(1)p Fr(;)g(:)g(:)g(:)f(;)h(q)2707 5089 y Fq(k)2766 5077 y Fp(\000)k Ft(1)p Fp(g)456 5216 y Ft(w)n(e)27 b(pro)n(v)n(e)f(the)i(equalit)n(y)41 b Fp(I)1326 5179 y Fq(q)1356 5188 y Fi(k)q Fg(+1)1463 5179 y Fn(\000)p Fm(1)1576 5216 y Ft(=)36 b Fp(J)1749 5179 y Fn(0)1800 5216 y Ft(similarly)-7 b(.)1251 b Fe(\003)p eop %%Page: 6 6 6 5 bop 456 251 a Fm(6)1200 b(MICHAL)22 b(KUPSA)555 450 y Ft(F)-7 b(or)25 b(eac)n(h)g Fr(k)h Fp(\025)d Ft(1)i(w)n(e)h(ha)n(v)n (e)e(th)n(us)i(a)f(one-to-one)f(map)i Fr(\015)2299 462 y Fq(k)2363 450 y Ft(:)37 b Fp(L)2480 414 y Fq(q)2510 423 y Fi(k)2547 414 y Fn(\000)p Fm(1)2636 450 y Ft(\(\006)2728 462 y Fq(\013)2775 450 y Ft(\))24 b Fp(!)37 b(L)3008 414 y Fq(k)3049 450 y Ft(\()p Fr(X)3150 462 y Fq(\013)3197 450 y Ft(\))26 b(giv)n(en)456 550 y(b)n(y)31 b Fr(J)621 565 y Fq(\015)656 574 y Fi(k)692 565 y Fm(\()p Fq(u)p Fm(\))817 550 y Ft(=)e Fr(I)947 562 y Fq(u)991 550 y Ft(.)49 b(F)-7 b(or)31 b(the)h(corresp)r(onding)d(sym)n(b)r(olic)j (spaces)e(w)n(e)h(get)h(a)f(homeomorphism)456 650 y Fr(\015)c Ft(:)c(\006)632 662 y Fq(\013)703 650 y Fp(!)g Fr(X)878 662 y Fq(\013)946 650 y Ft(giv)n(en)d(b)n(y)g Fr(\015)5 b Ft(\()p Fr(x)p Ft(\)\()p Fr(k)s Ft(\))24 b(=)f Fr(\015)1688 662 y Fq(k)1729 650 y Ft(\()p Fr(x)p Ft(\()p Fr(q)1877 662 y Fq(k)1923 650 y Fp(\000)5 b Ft(1\)\).)34 b(The)21 b(lo)r(cal)f(return)g(rates,)i(as)e(w)n(ell)g(as)g(the)456 749 y(functions)29 b(of)h(the)g(recurrence)e(jumps)i(are)f(carried)f(o) n(v)n(er)g(to)h(the)h(space)f Fr(X)2851 761 y Fq(\013)2898 749 y Ft(.)43 b(By)29 b(the)h(abuse)456 849 y(of)e(notation)h(w)n(e)f (k)n(eep)h(for)f(them)i(the)f(same)f(sym)n(b)r(ols)h Fr(R)p 2213 862 64 4 v(;)p 2313 782 V 14 w(R)d Ft(:)f Fr(X)2519 861 y Fq(\013)2592 849 y Fp(!)g Ft([0)p Fr(;)14 b Fp(1)p Ft(],)29 b Fr(r)2997 861 y Fq(k)3063 849 y Ft(:)d Fr(X)3181 861 y Fq(\013)3253 849 y Fp(!)f Fk(N)t Ft(.)456 949 y(W)-7 b(e)28 b(no)n(w)f(obtain)g(a)g(recursiv)n(e)f(form)n(ula)h (for)g Fr(r)1915 961 y Fq(k)1956 949 y Ft(.)456 1105 y Fx(Prop)s(osition)j(3.)41 b Fh(F)-6 b(or)30 b Fr(x)23 b Fp(2)g Fr(X)1460 1117 y Fq(\013)1537 1105 y Fh(we)30 b(have)h Fr(r)1888 1117 y Fn(\000)p Fm(1)1978 1105 y Ft(\()p Fr(x)p Ft(\))24 b(=)f(0)29 b Fh(and)1233 1334 y Fr(r)1270 1346 y Fq(k)1311 1334 y Ft(\()p Fr(x)p Ft(\))24 b(=)f Fr(x)1581 1346 y Fq(k)q Fm(+1)1706 1334 y Fr(q)1743 1346 y Fq(k)1803 1334 y Ft(+)18 b Fr(r)1923 1346 y Fq(k)q Fn(\000)p Fm(1)2049 1334 y Ft(\()p Fr(x)p Ft(\))24 b(=)2313 1230 y Fq(k)2272 1255 y Fj(X)2274 1432 y Fq(j)s Fm(=0)2405 1334 y Fr(x)2452 1346 y Fq(j)s Fm(+1)2572 1334 y Fr(q)2609 1346 y Fq(j)2644 1334 y Fr(:)456 1579 y Fh(Pr)l(o)l(of.)43 b Ft(Assume)31 b Fr(y)h Fp(2)d Ft(\006)1244 1591 y Fq(\013)1292 1579 y Ft(,)j Fr(x)d Ft(=)g Fr(\015)5 b Ft(\()p Fr(y)s Ft(\))29 b Fp(2)h Fr(X)1856 1591 y Fq(\013)1934 1579 y Ft(and)h Fr(k)h Fp(\025)d Ft(0.)47 b(W)-7 b(e)32 b(sho)n(w)e(\014rst) h(that)h(if)f Fr(J)3220 1594 y Fq(x)p Fm(\()p Fq(k)q Fm(\))3380 1579 y Ft(=)456 1687 y(\()p Fp(\000)p Ft(1\))627 1657 y Fq(k)q Fn(\000)p Fm(1)752 1687 y Ft([)p Fp(h)p Fr(a)p Fp(i)p Fr(;)14 b Fp(h)p Fr(b)p Fp(i)p Ft([,)34 b(then)f Fr(r)1331 1699 y Fq(k)q Fn(\000)p Fm(1)1458 1687 y Ft(\()p Fr(x)p Ft(\))e(=)g Fr(b)21 b Ft(+)g Fr(q)1876 1699 y Fq(k)q Fn(\000)p Fm(1)2002 1687 y Ft(.)51 b(If)33 b Fr(x)2211 1699 y Fq(k)2283 1687 y Fr(<)e(a)2423 1699 y Fq(k)2464 1687 y Ft(,)i(then)g Fr(J)2760 1702 y Fq(x)p Fm(\()p Fq(k)q Fm(\))2921 1687 y Ft(=)e Fr(I)3053 1702 y Fq(y)r Fm(\()p Fq(a)p Fm(\))3181 1687 y Ft(.)51 b(Since)456 1787 y Fr(I)492 1802 y Fq(y)r Fm(\()p Fq(a)p Fn(\000)p Fm(1\))728 1787 y Fp(6)p Ft(=)23 b Fr(I)852 1802 y Fq(y)r Fm(\()p Fq(a)p Fm(\))980 1787 y Ft(,)e Fp(j)p Fr(I)1083 1802 y Fq(y)r Fm(\()p Fq(a)p Fn(\000)p Fm(1\))1297 1787 y Fp(j)i Fr(>)g(\021)1472 1799 y Fq(k)q Fn(\000)p Fm(1)1617 1787 y Ft(and)d Fr(r)1808 1799 y Fq(k)q Fn(\000)p Fm(1)1934 1787 y Ft(\()p Fr(x)p Ft(\))25 b(=)d Fr(a)h Ft(=)g Fr(b)s Ft(+)s Fr(q)2456 1799 y Fq(k)q Fn(\000)p Fm(1)2581 1787 y Ft(.)34 b(Let)20 b Fr(x)2826 1799 y Fq(k)2890 1787 y Ft(=)j Fr(a)3022 1799 y Fq(k)3063 1787 y Ft(.)34 b(Since)20 b(the)456 1902 y(form)27 b(of)h(partition)g Fp(f)p Fr(J)1182 1914 y Fq(u)1248 1902 y Fp(j)c Fr(u)f Fp(2)38 b(L)1516 1865 y Fq(k)q Fm(+1)1641 1902 y Ft(\()p Fr(X)1742 1914 y Fq(\013)1789 1902 y Ft(\))p Fr(;)14 b(u)p Ft(\()p Fr(k)s Ft(\))24 b(=)f Fr(x)p Ft(\()p Fr(k)s Ft(\))p Fp(g)28 b Ft(of)g Fr(J)2496 1917 y Fq(x)p Fm(\()p Fq(k)q Fm(\))2655 1902 y Ft(w)n(e)f(get)h Fr(J)2962 1917 y Fq(x)p Fm(\()p Fq(k)q Fm(\))3116 1902 y Ft(=)23 b Fr(I)3240 1914 y Fm(1)3297 1902 y Fp([)c Fr(I)3407 1914 y Fm(2)456 2001 y Ft(where)696 2148 y Fr(I)732 2160 y Fm(1)793 2148 y Ft(=)k(\()p Fp(\000)p Ft(1\))1052 2114 y Fq(k)q Fn(\000)p Fm(1)1177 2148 y Ft([)p Fp(h)p Fr(a)p Fp(i)p Fr(;)14 b Fp(h)p Fr(a)19 b Ft(+)f Fr(q)1560 2160 y Fq(k)1601 2148 y Fp(i)p Ft([)p Fr(;)98 b(I)1813 2160 y Fm(2)1873 2148 y Ft(=)23 b(\()p Fp(\000)p Ft(1\))2132 2114 y Fq(k)q Fn(\000)p Fm(1)2257 2148 y Ft([)p Fp(h)p Fr(a)c Ft(+)f Fr(q)2495 2160 y Fq(k)2536 2148 y Fp(i)p Fr(;)c Fp(h)p Fr(a)19 b Ft(+)f Fr(q)2820 2160 y Fq(k)2879 2148 y Fp(\000)g Fr(q)2999 2160 y Fq(k)q Fn(\000)p Fm(1)3125 2148 y Fp(i)p Ft([)p Fr(;)456 2296 y(I)492 2308 y Fm(1)529 2296 y Fr(;)c(I)602 2308 y Fm(2)665 2296 y Fp(2)40 b(I)811 2260 y Fq(a)p Fm(+)p Fq(q)928 2269 y Fi(k)969 2296 y Ft(,)29 b Fp(j)p Fr(I)1080 2308 y Fm(1)1118 2296 y Fp(j)d Ft(=)f Fr(\021)1298 2308 y Fq(k)1339 2296 y Ft(,)k Fp(j)p Fr(I)1450 2308 y Fm(2)1488 2296 y Fp(j)c Ft(=)g Fr(\021)1667 2308 y Fq(k)q Fn(\000)p Fm(1)1822 2296 y Ft(and)k Fr(I)2021 2311 y Fq(y)r Fm(\()p Fq(a)p Fm(+)p Fq(q)2200 2320 y Fi(k)2237 2311 y Fn(\000)p Fm(1\))2377 2296 y Ft(=)c Fr(J)2513 2311 y Fq(x)p Fm(\()p Fq(k)q Fm(\))2673 2296 y Ft(and)k(either)f Fr(I)3108 2311 y Fq(y)r Fm(\()p Fq(a)p Fm(+)p Fq(q)3287 2320 y Fi(k)3324 2311 y Fm(\))3380 2296 y Ft(=)456 2396 y Fr(I)492 2408 y Fm(1)562 2396 y Ft(or)33 b Fr(I)706 2411 y Fq(y)r Fm(\()p Fq(a)p Fm(+)p Fq(q)885 2420 y Fi(k)921 2411 y Fm(\))984 2396 y Ft(=)e Fr(I)1116 2408 y Fm(2)1154 2396 y Ft(.)53 b(Hence)33 b Fp(j)p Fr(I)1541 2411 y Fq(y)r Fm(\()p Fq(a)p Fm(+)p Fq(q)1720 2420 y Fi(k)1757 2411 y Fm(\))1787 2396 y Fp(j)f(\024)g Fr(\021)1980 2408 y Fq(k)q Fn(\000)p Fm(1)2106 2396 y Ft(,)j Fp(j)p Fr(I)2223 2411 y Fq(y)r Fm(\()p Fq(a)p Fm(+)p Fq(q)2402 2420 y Fi(k)2439 2411 y Fn(\000)p Fm(1\))2554 2396 y Fp(j)d Fr(>)f(\021)2746 2408 y Fq(k)q Fn(\000)p Fm(1)2906 2396 y Ft(and)h Fr(r)3109 2408 y Fq(k)q Fn(\000)p Fm(1)3236 2396 y Ft(\()p Fr(x)p Ft(\))h(=)456 2496 y Fr(a)18 b Ft(+)g Fr(q)638 2508 y Fq(k)702 2496 y Ft(=)23 b Fr(b)18 b Ft(+)g Fr(q)964 2508 y Fq(k)q Fn(\000)p Fm(1)1090 2496 y Ft(.)555 2596 y(Assume)j(no)n(w)f(that)g Fr(J)1240 2611 y Fq(x)p Fm(\()p Fq(k)q Fm(+1\))1478 2596 y Ft(=)j(\()p Fp(\000)p Ft(1\))1737 2565 y Fq(k)1777 2596 y Ft([)p Fp(h)p Fr(c)p Fp(i)p Fr(;)14 b Fp(h)p Fr(d)p Fp(i)p Ft([,)23 b(so)d Fr(r)2245 2608 y Fq(k)2287 2596 y Ft(\()p Fr(x)p Ft(\))k(=)e Fr(d)t Ft(+)t Fr(q)2662 2608 y Fq(k)2704 2596 y Ft(.)34 b(Put)21 b(j=1)f(if)h Fr(x)3185 2608 y Fq(k)3249 2596 y Ft(=)i Fr(a)3381 2608 y Fq(k)3421 2596 y Ft(,)456 2696 y(j=0)i(otherwise.)35 b(It)26 b(follo)n(ws)f Fr(d)e Ft(=)g Fr(a)15 b Ft(+)f(\()p Fr(x)1744 2708 y Fq(k)q Fm(+1)1870 2696 y Ft(\))h Fp(\000)f Ft(1\))p Fr(q)2107 2708 y Fq(k)2163 2696 y Ft(+)g Fr(j)5 b(q)2318 2708 y Fq(k)2385 2696 y Ft(and)25 b Fr(a)e Ft(=)g Fr(b)14 b Ft(+)h Fr(q)2866 2708 y Fq(k)q Fn(\000)p Fm(1)3006 2696 y Fp(\000)g Fr(j)5 b(q)3162 2708 y Fq(k)3202 2696 y Ft(.)37 b(Th)n(us)569 2842 y Fr(r)606 2854 y Fq(k)647 2842 y Ft(\()p Fr(x)p Ft(\))20 b Fp(\000)e Fr(r)898 2854 y Fq(k)q Fn(\000)p Fm(1)1024 2842 y Ft(\()p Fr(x)p Ft(\))84 b(=)e(\()p Fr(d)19 b Ft(+)f Fr(q)1580 2854 y Fq(k)1621 2842 y Ft(\))h Fp(\000)f Ft(\()p Fr(b)g Ft(+)g Fr(q)1961 2854 y Fq(k)q Fn(\000)p Fm(1)2087 2842 y Ft(\))24 b(=)e(\()p Fr(a)d Ft(+)f Fr(x)2455 2854 y Fq(j)s Fm(+1)2574 2842 y Fr(q)2611 2854 y Fq(k)2671 2842 y Ft(+)g Fr(j)5 b(q)2830 2854 y Fq(k)2871 2842 y Ft(\))18 b Fp(\000)g Ft(\()p Fr(a)h Ft(+)f Fr(j)5 b(q)3258 2854 y Fq(k)3299 2842 y Ft(\))1219 2967 y(=)82 b Fr(x)1413 2979 y Fq(k)q Fm(+1)1539 2967 y Fr(q)1576 2979 y Fq(k)1617 2967 y Fr(:)3380 3114 y Fe(\003)456 3281 y Fx(Prop)s(osition)30 b(4.)41 b Fh(F)-6 b(or)30 b(every)g Fr(x)24 b Fp(2)f Fr(X)1680 3293 y Fq(\013)1757 3281 y Fh(we)30 b(have)h Fr(q)2108 3293 y Fq(k)2172 3281 y Fp(\024)22 b Fr(r)2296 3293 y Fq(k)2338 3281 y Ft(\()p Fr(x)p Ft(\))i Fp(\024)e Fr(q)2597 3293 y Fq(k)q Fm(+1)2741 3281 y Ft(+)c Fr(q)2861 3293 y Fq(k)2920 3281 y Fp(\000)g Ft(1)p Fh(.)456 3437 y(Pr)l(o)l(of.)43 b Ft(Clearly)35 b Fr(q)1050 3449 y Fn(\000)p Fm(1)1176 3437 y Ft(=)h(0)h(=)g Fr(r)1495 3449 y Fn(\000)p Fm(1)1584 3437 y Ft(\()p Fr(x)p Ft(\))h(=)f(0)f(=)h Fr(q)2052 3449 y Fm(0)2113 3437 y Ft(+)24 b Fr(q)2239 3449 y Fn(\000)p Fm(1)2352 3437 y Fp(\000)g Ft(1,)37 b Fr(q)2580 3449 y Fm(1)2655 3437 y Ft(=)f(1)h Fp(\024)f Fr(r)2973 3449 y Fm(1)3011 3437 y Ft(\()p Fr(x)p Ft(\))i Fp(\024)f Fr(a)3306 3449 y Fm(1)3380 3437 y Ft(=)456 3537 y Fr(q)493 3549 y Fm(1)548 3537 y Ft(+)18 b Fr(q)668 3549 y Fm(0)724 3537 y Fp(\000)g Ft(1.)37 b(Assume)27 b(that)h(the)g(statemen)n(t)g(holds)f(for)g(all)g (in)n(tegers)g(less)g(than)h Fr(k)s Ft(.)36 b(Then)716 3684 y Fr(r)753 3696 y Fq(k)795 3684 y Ft(\()p Fr(x)p Ft(\))24 b(=)e Fr(x)1064 3696 y Fq(k)q Fm(+1)1190 3684 y Fr(q)1227 3696 y Fq(k)1286 3684 y Ft(+)c Fr(r)1406 3696 y Fq(k)q Fn(\000)p Fm(1)1532 3684 y Ft(\()p Fr(x)p Ft(\))25 b Fp(\024)d Fr(a)1799 3696 y Fq(k)q Fm(+1)1924 3684 y Fr(q)1961 3696 y Fq(k)2020 3684 y Ft(+)c Fr(q)2140 3696 y Fq(k)2200 3684 y Ft(+)g Fr(q)2320 3696 y Fq(k)q Fn(\000)p Fm(1)2464 3684 y Fp(\000)g Ft(1)23 b(=)f Fr(q)2736 3696 y Fq(k)2796 3684 y Ft(+)c Fr(q)2916 3696 y Fq(k)q Fm(+1)3059 3684 y Fp(\000)g Ft(1)456 3830 y(If)28 b Fr(x)586 3842 y Fq(k)q Fm(+1)734 3830 y Fp(\025)23 b Ft(1,)k(then)h Fr(r)1140 3842 y Fq(k)1181 3830 y Ft(\()p Fr(x)p Ft(\))c(=)f Fr(x)1451 3842 y Fq(k)q Fm(+1)1576 3830 y Fr(q)1613 3842 y Fq(k)1673 3830 y Ft(+)18 b Fr(r)1793 3842 y Fq(k)q Fn(\000)p Fm(1)1919 3830 y Ft(\()p Fr(x)p Ft(\))24 b Fp(\025)f Fr(q)2179 3842 y Fq(k)2220 3830 y Ft(.)37 b(If)28 b Fr(x)2410 3842 y Fq(k)q Fm(+1)2558 3830 y Ft(=)23 b(0,)k(then)h Fr(x)2974 3842 y Fq(k)3038 3830 y Ft(=)23 b Fr(a)3170 3842 y Fq(k)3238 3830 y Ft(and)877 3977 y Fr(r)914 3989 y Fq(k)955 3977 y Ft(\()p Fr(x)p Ft(\))h(=)f Fr(r)1215 3989 y Fq(k)q Fn(\000)p Fm(1)1341 3977 y Ft(\()p Fr(x)p Ft(\))i(=)d Fr(a)1608 3989 y Fq(k)1649 3977 y Fr(q)1686 3989 y Fq(k)q Fn(\000)p Fm(1)1830 3977 y Ft(+)c Fr(r)1950 3989 y Fq(k)q Fn(\000)p Fm(2)2077 3977 y Ft(\()p Fr(x)p Ft(\))24 b Fp(\025)e Fr(a)2343 3989 y Fq(k)2384 3977 y Fr(q)2421 3989 y Fq(k)q Fn(\000)p Fm(1)2565 3977 y Ft(+)c Fr(q)2685 3989 y Fq(k)q Fn(\000)p Fm(2)2834 3977 y Ft(=)23 b Fr(q)2959 3989 y Fq(k)3000 3977 y Fr(:)3380 4124 y Fe(\003)456 4291 y Fx(Prop)s(osition)30 b(5.)41 b Fh(De\014ne)29 b(the)h(p)l(oints)g Fr(b;)14 b(c;)g(d)22 b Fp(2)i Fr(X)2091 4303 y Fq(\013)2167 4291 y Fh(by)636 4438 y Fr(b)e Ft(=)h(\()p Fr(a)858 4450 y Fm(1)895 4438 y Fr(;)14 b(a)976 4450 y Fm(2)1013 4438 y Fr(;)g(a)1094 4450 y Fm(3)1131 4438 y Fr(;)g(:)g(:)g(:)p Ft(\))p Fr(;)99 b(c)23 b Ft(=)g(\(1)p Fr(;)14 b(a)1721 4450 y Fm(2)1758 4438 y Fr(;)g Ft(0)p Fr(;)g(a)1918 4450 y Fm(4)1954 4438 y Fr(;)g Ft(0)p Fr(;)g(a)2114 4450 y Fm(6)2150 4438 y Fr(;)g(:)g(:)g(:)p Ft(\))p Fr(;)99 b(d)24 b Ft(=)e(\()p Fr(a)2668 4450 y Fm(1)2706 4438 y Fr(;)14 b Ft(0)p Fr(;)g(a)2866 4450 y Fm(3)2902 4438 y Fr(;)g 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Fq(k)q Fm(+1)p 2457 5118 162 4 v 2499 5194 a Fr(q)2536 5206 y Fq(k)2675 5137 y Ft(=)46 b Fx(r)2825 5149 y Fm(1)2863 5137 y Fr(:)p eop %%Page: 7 7 7 6 bop 1173 251 a Fm(LOCAL)29 b(RETURN)h(RA)-5 b(TES)28 b(IN)i(STURMIAN)f(SHIFTS)683 b(7)456 450 y Fh(Pr)l(o)l(of.)43 b Ft(It)28 b(is)f(easy)g(to)g(see)h(that)f(for)g Fr(k)f Fp(2)e Fk(N)t Ft(,)1491 678 y Fr(r)1528 690 y Fq(k)1569 678 y Ft(\()p Fr(b)p Ft(\))83 b(=)1942 574 y Fq(k)1900 599 y Fj(X)1903 776 y Fq(j)s Fm(=0)2034 678 y Fr(a)2078 690 y Fq(j)s Fm(+1)2197 678 y Fr(q)2234 690 y Fq(j)2292 678 y Ft(=)22 b Fr(q)2416 690 y Fq(k)q Fm(+1)2560 678 y Ft(+)c Fr(q)2680 690 y Fq(k)2739 678 y Fp(\000)g Ft(1)1050 981 y Fr(r)1087 993 y Fm(2)p Fq(k)q Fn(\000)p Fm(1)1246 981 y Ft(\()p Fr(c)p Ft(\))24 b(=)e Fr(r)1494 993 y Fm(2)p Fq(k)1569 981 y Ft(\()p Fr(c)p Ft(\))83 b(=)g(1)18 b(+)2085 877 y Fq(k)2043 902 y Fj(X)2045 1079 y Fq(j)s Fm(=1)2177 981 y Fr(a)2221 993 y Fm(2)p Fq(j)2289 981 y Fr(q)2326 993 y Fm(2)p Fq(j)s Fn(\000)p Fm(1)2502 981 y Ft(=)23 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Fj(\022)1860 2453 y Ft(lim)14 b(sup)1903 2523 y Fq(k)q Fn(!1)2138 2396 y Fr(q)2175 2408 y Fm(2)p Fq(k)q Fm(+1)p 2138 2433 196 4 v 2180 2509 a Fr(q)2217 2521 y Fm(2)p Fq(k)2343 2453 y Fr(;)g Ft(lim)g(sup)2423 2523 y Fq(k)q Fn(!1)2700 2396 y Fr(q)2737 2408 y Fm(2)p Fq(k)p 2658 2433 V 2658 2509 a Fr(q)2695 2521 y Fm(2)p Fq(k)q Fn(\000)p Fm(1)2864 2335 y Fj(\023)1483 2657 y Fp(\025)83 b Ft(lim)14 b(sup)1673 2727 y Fq(k)q Fn(!1)1909 2600 y Fr(q)1946 2612 y Fq(k)q Fm(+1)p 1909 2637 162 4 v 1951 2713 a Fr(q)1988 2725 y Fq(k)2104 2657 y Ft(=)22 b Fx(r)2230 2669 y Fm(1)2268 2657 y Fr(:)3380 2850 y Fe(\003)555 3022 y Ft(W)-7 b(e)29 b(ha)n(v)n(e)e(not)h(b)r(een)h(able)f(to)g(obtain)g(a)g(form)n(ula)g (for)g(min)p 2429 2956 64 4 v 14 w Fr(R)p Ft(.)40 b(Our)27 b(results,)h(ho)n(w)n(ev)n(er)e(are)456 3122 y(su\016cien)n(t)32 b(to)g(get)g(form)n(ulas)f(for)g Fx(r)1570 3134 y Fm(0)1640 3122 y Ft(and)h Fx(r)1845 3134 y Fm(1)1883 3122 y Ft(.)50 b(No)n(w,)33 b(put)g(some)f(b)r(ounds)g(for)g(the)g(v)-5 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y Fm(0)1725 3793 y Fp(\024)g Fr(\015)1860 3758 y Fn(\000)p Fm(2)1972 3793 y Fr(<)h(\015)28 b Fp(\024)22 b Fx(r)2257 3805 y Fm(1)2318 3793 y Fp(\024)g Fr(M)27 b Ft(+)18 b(1)456 3983 y Fh(Mor)l(e)l(over,)31 b Fx(r)885 3995 y Fm(1)946 3983 y Ft(=)22 b Fr(\015)5 b Fh(\()30 b(r)l(esp.)39 b Fx(r)1393 3995 y Fm(0)1453 3983 y Ft(=)23 b Fr(\015)1589 3953 y Fn(\000)p Fm(2)1678 3983 y Fh(\))29 b(if)i(and)f(only)h(if)f Fr(M)i Ft(=)22 b(1)p Fh(.)456 4158 y(Pr)l(o)l(of.)43 b Ft(Let)37 b Fr(\013)j Ft(=)g([0)p Fr(;)14 b(a)1218 4170 y Fm(1)1254 4158 y Fr(;)g(a)1335 4170 y Fm(2)1372 4158 y Fr(;)g(::)p Ft(])38 b(b)r(e)g(irrational,)g Fr(M)49 b Ft(=)39 b Fr(l)r(imsup)c(a)2610 4170 y Fq(k)2651 4158 y Ft(,)40 b Fr(\015)45 b Ft(=)2916 4077 y Fn(p)p 2970 4077 34 3 v 2970 4125 a Fm(5+1)p 2916 4139 172 4 v 2985 4186 a(2)3097 4158 y Ft(.)67 b(Denote)456 4261 y Fr(B)519 4273 y Fq(k)582 4261 y Ft(=)680 4220 y Fq(q)710 4229 y Fi(k)q Fg(+1)p 680 4242 138 4 v 716 4290 a Fq(q)746 4299 y Fi(k)828 4261 y Ft(.)37 b(Then)27 b Fr(a)1148 4273 y Fq(k)1212 4261 y Fp(\024)c Fr(B)1363 4273 y Fq(k)1427 4261 y Fp(\024)f Fr(a)1558 4273 y Fq(k)q Fm(+1)1702 4261 y Ft(+)c(1)27 b(and)g Fx(r)2054 4273 y Fm(1)2115 4261 y Ft(=)c(lim)14 b(sup)f Fr(B)2533 4273 y Fq(k)2574 4261 y Ft(,)1196 4470 y Fx(r)1235 4482 y Fm(0)1296 4470 y Ft(=)23 b(lim)14 b(inf)1834 4414 y(1)p 1637 4451 435 4 v 1637 4531 a Fr(B)1700 4543 y Fq(k)1760 4531 y Ft(+)k(1)g Fp(\000)2013 4498 y Fm(1)p 1996 4512 67 4 v 1996 4559 a Fq(q)2026 4568 y Fi(k)2105 4470 y Ft(=)23 b(lim)14 b(inf)2550 4414 y(1)p 2447 4451 247 4 v 2447 4527 a Fr(B)2510 4539 y Fq(k)2569 4527 y Ft(+)k(1)555 4692 y(If)37 b(the)g(con)n(tin)n (ued)f(fraction)f(expansion)h(of)g Fr(\013)h Ft(is)f(un)n(b)r(ounded,)j (then)e(also)e Fp(f)p Fr(B)3122 4704 y Fq(k)3163 4692 y Fp(g)3205 4662 y Fn(1)3205 4715 y Fq(k)q Fm(=0)3366 4692 y Ft(is.)456 4792 y(Hence)27 b Fx(r)741 4804 y Fm(0)802 4792 y Ft(=)c(0)k(and)g Fx(r)1159 4804 y Fm(1)1220 4792 y Ft(=)c Fp(1)p Ft(.)555 4891 y(Let)31 b Fr(M)37 b Fp(2)29 b Fk(N)t Ft(.)52 b(Then)31 b Fr(M)37 b Fp(\024)28 b Fx(r)1508 4903 y Fm(0)1573 4891 y Fp(\024)g Fr(M)h Ft(+)20 b(1.)46 b(If)31 b Fr(M)37 b Fp(\025)28 b Ft(2,)j(then)h Fr(\015)g(<)c(M)37 b Fp(\024)28 b Fx(r)2976 4903 y Fm(0)3014 4891 y Ft(.)46 b(If)31 b Fr(M)37 b Ft(=)28 b(1,)456 4991 y(then)g(there)f(exists)g Fr(n)1136 5003 y Fm(0)1196 4991 y Fp(2)d Fk(N)t Ft(,)34 b(suc)n(h)27 b(that)h(for)f(ev)n(ery)f Fr(n)d(>)g(n)2310 5003 y Fm(0)2347 4991 y Ft(,)k Fr(a)2441 5003 y Fq(n)2510 4991 y Ft(=)22 b(1.)37 b(Hence)862 5176 y(lim)14 b(sup)g Fr(B)1193 5188 y Fq(k)1257 5176 y Ft(=)23 b(1)18 b(+)1651 5120 y(1)p 1498 5157 348 4 v 1498 5233 a(lim)c(inf)21 b Fr(B)1805 5245 y Fq(k)1938 5176 y Fr(;)84 b Ft(lim)14 b(inf)21 b Fr(B)2352 5188 y Fq(k)2415 5176 y Ft(=)i(1)18 b(+)2821 5120 y(1)p 2656 5157 372 4 v 2656 5233 a(lim)c(sup)g Fr(B)2987 5245 y Fq(k)p eop %%Page: 8 8 8 7 bop 456 251 a Fm(8)1200 b(MICHAL)22 b(KUPSA)456 450 y Ft(It)k(implies)f(that)h Fx(r)1040 462 y Fm(0)1101 450 y Ft(=)d(1)14 b(+)1398 417 y Fm(1)p 1334 431 163 4 v 1334 483 a(1+)1443 461 y Fg(1)p 1428 470 59 3 v 1428 503 a Fb(r)1454 515 y Fg(0)1506 450 y Ft(.)36 b(This)26 b(equalit)n(y)f(ha)n(v)n(e)g(just)h(one)f(p)r(ositiv)n(e)g(solution)g Fx(r)3225 462 y Fm(0)3286 450 y Ft(=)e Fr(\015)5 b Ft(.)456 594 y(All)28 b(prop)r(erties)e(of)i Fx(r)1116 606 y Fm(1)1181 594 y Ft(is)g(giv)n(en)e(b)n(y)i(the)g(equalit)n(y)f Fx(r)2095 606 y Fm(0)2155 594 y Ft(=)2310 562 y Fm(1)p 2253 576 148 4 v 2253 623 a Fc(r)2284 631 y Fg(1)2316 623 y Fm(+1)2410 594 y Ft(.)947 b Fe(\003)1628 841 y Ft(4.)41 b Fs(The)32 b(measure)555 991 y Ft(W)-7 b(e)35 b(are)f(going)g(to)g(sho)n(w)g(that)h(the)h(constan)n(ts)d Fx(r)2152 1003 y Fm(0)2225 991 y Ft(and)i Fx(r)2433 1003 y Fm(1)2505 991 y Ft(are)f(assumed)g(b)n(y)h Fr(R)p 3114 1004 64 4 v 35 w Ft(and)p 3381 924 V 35 w Fr(R)456 1091 y Ft(almost)c(ev)n(erywhere.)47 b(The)32 b(unique)f(in)n(v)-5 b(arian)n(t)31 b(measure)g Fr(\026)g Ft(on)h(\006)2594 1103 y Fq(\013)2673 1091 y Ft(is)f(carried)g(o)n(v)n(er)f(to)h(the)456 1192 y(space)g Fr(X)751 1204 y Fq(\013)829 1192 y Ft(using)h(the)g (length)g(of)f(asso)r(ciated)f(in)n(terv)-5 b(als.)49 b(If)32 b Fr(u)d Fp(2)44 b(L)2652 1155 y Fq(k)2692 1192 y Ft(\()p Fr(X)2793 1204 y Fq(\013)2841 1192 y Ft(\),)33 b(then)f(the)g(mea-)456 1291 y(sure)i(of)h(the)g(cylinder)g(of)g Fr(u)f Ft(is)h Fr(\026)p Fp(f)p Fr(x)h Fp(2)f Fr(X)1824 1303 y Fq(\013)1871 1291 y Fp(j)p Fr(x)p Ft(\()p Fr(k)s Ft(\))i(=)e Fr(u)p Fp(g)f Ft(=)h Fp(j)p Fr(J)2481 1303 y Fq(u)2525 1291 y Fp(j)p Ft(.)59 b(De\014ne)35 b(the)h(pro)5 b(jections)456 1391 y Fr(W)534 1403 y Fq(k)612 1391 y Ft(:)36 b Fr(X)740 1403 y Fq(\013)824 1391 y Fp(!)h(f)p Ft(0)p Fr(;)14 b Ft(1)p Fr(;)g(:)g(:)g(:)e(;)i(a)1334 1403 y Fq(k)1375 1391 y Fp(g)35 b Ft(b)n(y)h Fr(W)1654 1403 y Fq(k)1695 1391 y Ft(\()p Fr(x)p Ft(\))i(=)e Fr(x)1992 1403 y Fq(k)2034 1391 y Ft(.)61 b(Then)36 b Fr(W)2421 1403 y Fq(k)2498 1391 y Ft(are)f(random)g(v)-5 b(ariables)35 b(and)456 1490 y(\()p Fr(W)566 1502 y Fq(k)607 1490 y Ft(\))639 1502 y Fq(k)q Fn(\025)p Fm(1)791 1490 y Ft(is)26 b(a)f(nonstationary)f(Mark)n(o)n(v)f(c)n(hain.)36 b(Using)26 b(Prop)r(osition)e Fx(??)i Ft(w)n(e)f(get)g(the)h(transi-)456 1590 y(tion)h(probabilities.)1313 1745 y Fr(\026)p Ft([)p Fr(W)1464 1757 y Fm(1)1525 1745 y Ft(=)c Fr(j)5 b Ft(])83 b(=)f Fr(\021)1946 1757 y Fm(0)2007 1745 y Ft(=)23 b Fr(\013)211 b Ft(for)82 b(1)23 b Fp(\024)g Fr(j)28 b(<)22 b(a)2887 1757 y Fm(1)1313 1880 y Fr(\026)p Ft([)p Fr(W)1464 1892 y Fm(1)1525 1880 y Ft(=)h Fr(j)5 b Ft(])83 b(=)f Fr(\021)1946 1892 y Fm(0)2002 1880 y Ft(+)18 b Fr(\021)2126 1892 y Fm(1)2359 1880 y Ft(for)82 b Fr(j)28 b Ft(=)23 b Fr(a)2735 1892 y Fm(1)888 2037 y Fr(\026)p Ft([)p Fr(W)1039 2049 y Fq(k)q Fm(+1)1187 2037 y Ft(=)g Fr(j)5 b Fp(j)p Fr(W)1415 2049 y Fq(k)1479 2037 y Fr(<)23 b(a)1611 2049 y Fq(k)1652 2037 y Ft(])83 b(=)1958 1980 y Fr(\021)1999 1992 y Fq(k)p 1915 2018 168 4 v 1915 2094 a Fr(\021)1956 2106 y Fq(k)q Fn(\000)p Fm(1)2359 2037 y Ft(for)f(1)23 b Fp(\024)g Fr(j)28 b(<)22 b(a)2887 2049 y Fq(k)q Fm(+1)888 2252 y Fr(\026)p Ft([)p Fr(W)1039 2264 y Fq(k)q Fm(+1)1187 2252 y Ft(=)h Fr(j)5 b Fp(j)p Fr(W)1415 2264 y Fq(k)1479 2252 y Fr(<)23 b(a)1611 2264 y Fq(k)1652 2252 y Ft(])83 b(=)1915 2196 y Fr(\021)1956 2208 y Fq(k)2016 2196 y Ft(+)18 b Fr(\021)2140 2208 y Fq(k)q Fm(+1)p 1915 2233 350 4 v 2007 2309 a Fr(\021)2048 2321 y Fq(k)q Fn(\000)p Fm(1)2359 2252 y Ft(for)82 b Fr(j)28 b Ft(=)23 b Fr(a)2735 2264 y Fq(k)q Fm(+1)888 2456 y Fr(\026)p Ft([)p Fr(W)1039 2468 y Fq(k)q Fm(+1)1187 2456 y Ft(=)g Fr(j)5 b Fp(j)p Fr(W)1415 2468 y Fq(k)1479 2456 y Ft(=)23 b Fr(a)1611 2468 y Fq(k)1652 2456 y Ft(])83 b(=)2050 2399 y Fr(\021)2091 2411 y Fq(k)p 1915 2436 351 4 v 1915 2512 a Fr(\021)1956 2524 y Fq(k)q Fn(\000)p Fm(1)2101 2512 y Ft(+)18 b Fr(\021)2225 2524 y Fq(k)2359 2456 y Ft(for)82 b(0)23 b Fp(\024)g Fr(j)28 b(<)22 b(a)2887 2468 y Fq(k)q Fm(+1)888 2671 y Fr(\026)p Ft([)p Fr(W)1039 2683 y Fq(k)q Fm(+1)1187 2671 y Ft(=)h Fr(j)5 b Fp(j)p Fr(W)1415 2683 y Fq(k)1479 2671 y Ft(=)23 b Fr(a)1611 2683 y Fq(k)1652 2671 y Ft(])83 b(=)1916 2615 y Fr(\021)1957 2627 y Fq(k)2016 2615 y Ft(+)18 b Fr(\021)2140 2627 y Fq(k)q Fm(+1)p 1915 2652 V 1915 2728 a Fr(\021)1956 2740 y Fq(k)q Fn(\000)p Fm(1)2101 2728 y Ft(+)g Fr(\021)2225 2740 y Fq(k)2359 2671 y Ft(for)82 b Fr(j)28 b Ft(=)23 b Fr(a)2735 2683 y Fq(k)q Fm(+1)456 2883 y Fx(Theorem)41 b(7.)46 b Fh(If)39 b(the)f(c)l(ontinue)l(d)g(fr)l (action)i(exp)l(ansion)e(is)h(unb)l(ounde)l(d,)i(then)d Fr(R)p 3060 2896 64 4 v 1 w Ft(\()p Fr(x)p Ft(\))h(=)f(0)p Fh(,)p 456 2919 V 456 2986 a Fr(R)p Ft(\()p Fr(x)p Ft(\))24 b(=)f Fp(1)30 b Fh(almost)g(everywher)l(e.)456 3150 y(Pr)l(o)l(of.)43 b Ft(F)-7 b(or)27 b(ev)n(ery)f Fr(x)e Fp(2)f Fr(X)1302 3162 y Fq(\013)1377 3150 y Ft(w)n(e)k(ha)n(v)n(e)1325 3354 y Fr(x)1372 3366 y Fq(k)q Fm(+1)1520 3354 y Fp(\024)1618 3298 y Fr(x)1665 3310 y Fq(k)q Fm(+1)1790 3298 y Fr(q)1827 3310 y Fq(k)1886 3298 y Ft(+)18 b Fr(r)2006 3310 y Fq(k)q Fn(\000)p Fm(1)2133 3298 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Fq(k)2426 5190 y Fi(i)2480 5170 y Fp(\024)g Fr(m;)14 b Ft(1)22 b Fp(\024)h Fr(i)g Fp(\024)f Fr(n)p Fp(g)h(\024)f Ft(2)3213 5136 y Fn(\000)p Fq(n)p Fm(+1)p eop %%Page: 9 9 9 8 bop 1173 251 a Fm(LOCAL)29 b(RETURN)h(RA)-5 b(TES)28 b(IN)i(STURMIAN)f(SHIFTS)683 b(9)456 450 y Ft(so)33 b Fr(\026)p Fp(f)p Fr(x)i Fp(2)g Fr(X)897 462 y Fq(\013)944 450 y Fp(j)p Fr(R)p 967 463 64 4 v Ft(\()p Fr(x)p Ft(\))h Fr(>)1299 417 y Fm(1)p 1286 431 59 4 v 1286 479 a Fq(m)1355 450 y Fp(g)e Ft(=)g(0)g(and)g Fr(R)p 1774 463 64 4 v 1 w Ft(\()p Fr(x)p Ft(\))h(=)f(0)g(almost)g(ev)n(erywhere.)56 b(W)-7 b(e)34 b(pro)n(v)n(e)f(no)n(w)456 560 y(the)c(statemen)n(t)g (for)p 1115 494 V 28 w Fr(R)q Ft(.)41 b(Giv)n(en)29 b Fr(")c Fp(2)p Ft(]0)p Fr(;)14 b Ft(1[,)29 b(let)g Fr(m)g Ft(b)r(e)g(an)g(in)n(teger)f(with)i(1)18 b Fp(\000)h Fr(")g Ft(+)3053 528 y Fm(4)p 3040 542 59 4 v 3040 589 a Fq(m)3135 560 y Ft(=)25 b Fr(\016)j(<)d Ft(1.)456 660 y(Assume)i(that)h Fr(k)21 b Ft(+)d(1)23 b Fp(2)g Fr(C)1292 672 y Fq(m)1383 660 y Ft(and)28 b(let)g Fr(x)23 b Fp(2)h Fr(X)1883 672 y Fq(\013)1958 660 y Ft(b)r(e)k(suc)n(h)f(that)h Fr(r)2475 672 y Fq(k)2516 660 y Ft(\()p Fr(x)p Ft(\))p Fr(=q)2706 672 y Fq(k)q Fm(+1)2855 660 y Fp(\025)22 b Fr(")p Ft(.)37 b(Then)923 871 y Fr(")83 b Fp(\024)1203 815 y Fr(x)1250 827 y Fq(k)q Fm(+1)1375 815 y Fr(q)1412 827 y Fq(k)1471 815 y Ft(+)18 b Fr(r)1591 827 y Fq(k)q Fn(\000)p Fm(1)1718 815 y Ft(\()p Fr(x)p Ft(\))p 1203 852 628 4 v 1261 928 a Fr(a)1305 940 y Fq(k)q Fm(+1)1429 928 y Fr(q)1466 940 y Fq(k)1526 928 y Ft(+)g Fr(q)1646 940 y Fq(k)q Fn(\000)p Fm(1)1863 871 y Fp(\024)1960 815 y Fr(x)2007 827 y Fq(k)q Fm(+1)2133 815 y Fr(q)2170 827 y Fq(k)2229 815 y Ft(+)g Fr(q)2349 827 y Fq(k)2408 815 y Ft(+)g Fr(q)2528 827 y Fq(k)q Fn(\000)p Fm(1)p 1960 852 695 4 v 2184 928 a Fr(a)2228 940 y Fq(k)q Fm(+1)2353 928 y Fr(q)2390 940 y Fq(k)2687 871 y Fp(\024)2785 815 y Fr(x)2832 827 y Fq(k)q Fm(+1)2976 815 y Ft(+)g(2)p 2785 852 316 4 v 2858 928 a Fr(a)2902 940 y Fq(k)q Fm(+1)790 1034 y 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4161 y Fq(n)2483 4170 y Fi(k)2520 4161 y Fn(\000)p Fq(m)p Fm(+1)2742 4149 y Ft(=)22 b Fr(b)2865 4161 y Fq(n)2906 4170 y Fi(k)2942 4161 y Fn(\000)p Fq(m)p Fm(+1)3141 4149 y Fp(g)p Fr(:)456 4312 y Ft(and)27 b Fr(D)e Ft(=)798 4250 y Fj(T)868 4271 y Fn(1)868 4337 y Fq(j)s Fm(=1)1001 4250 y Fj(S)1070 4271 y Fn(1)1070 4337 y Fq(k)q Fm(=)p Fq(j)1206 4312 y Fr(D)1275 4324 y Fq(k)1316 4312 y Ft(.)37 b(W)-7 b(e)27 b(sho)n(w)g Fr(\026)p Ft(\()p Fr(D)r Ft(\))d(=)e(1.)37 b(Let)27 b Fr(M)36 b Ft(b)r(e)28 b(a)f(b)r(ound)g(for)g(the)h(con)n (tin)n(ued)456 4415 y(fraction)e(expansion,)h(so)g Fr(a)1317 4427 y Fq(k)1381 4415 y Fp(\024)22 b Fr(M)36 b Ft(for)27 b(ev)n(ery)g Fr(k)s Ft(.)36 b(Then)28 b(for)f(an)n(y)g Fr(i)22 b Fp(\024)h Fr(a)2721 4427 y Fq(k)2762 4415 y Ft(,)28 b Fr(j)g Fp(\024)22 b Fr(a)3006 4427 y Fq(k)q Fm(+1)3159 4415 y Ft(w)n(e)27 b(ha)n(v)n(e)942 4616 y Fr(\026)p Ft([)p Fr(W)1093 4628 y Fq(k)q Fm(+1)1241 4616 y Ft(=)c Fr(j)5 b Fp(j)p Fr(W)1469 4628 y Fq(k)1533 4616 y Ft(=)23 b Fr(i)p 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Fj(1)2274 5103 y(A)2370 5120 y Ft(=)23 b(1)18 b Fp(\000)g Ft(0)k(=)h(1)p Fr(:)p eop %%Page: 10 10 10 9 bop 456 251 a Fm(10)1167 b(MICHAL)22 b(KUPSA)456 450 y Ft(Giv)n(en)34 b Fr(x)h Fp(2)g Fr(D)r Ft(,)i(there)d(exists)g(an) g(increasing)f(in)n(teger)h(sequence)g Fp(f)p Fr(k)2696 462 y Fq(j)2731 450 y Fp(g)2773 420 y Fn(1)2773 472 y Fq(j)s Fm(=1)2926 450 y Ft(suc)n(h)g(that)h Fr(x)g Fp(2)456 550 y Fr(D)525 562 y Fq(n)566 571 y Fi(k)597 584 y(j)636 550 y Ft(.)i(F)-7 b(or)27 b(eac)n(h)g Fr(j)5 b Ft(,)28 b(w)n(e)f(ha)n(v)n(e)456 815 y Fr(r)493 827 y Fq(n)534 836 y Fi(k)565 849 y(j)605 815 y Ft(\()p Fr(b)p Ft(\))19 b Fp(\000)f Fr(r)844 827 y Fq(n)885 836 y Fi(k)916 849 y(j)956 815 y Ft(\()p Fr(x)p Ft(\))84 b(=)1305 688 y Fq(n)1346 697 y Fi(k)1377 710 y(j)1298 736 y Fj(X)1305 913 y Fq(i)p Fm(=0)1432 815 y Fr(b)1468 827 y Fq(i)p Fm(+1)1580 815 y Fr(q)1617 827 y Fq(i)1663 815 y Fp(\000)1752 688 y Fq(n)1793 697 y Fi(k)1824 710 y(j)1746 736 y Fj(X)1752 913 y Fq(i)p Fm(=0)1880 815 y Fr(x)1927 827 y Fq(i)p Fm(+1)2039 815 y Fr(q)2076 827 y Fq(i)2126 815 y Ft(=)2214 688 y Fq(n)2255 697 y Fi(k)2286 710 y(j)2322 688 y Fn(\000)p Fq(m)2263 736 y Fj(X)2270 913 y Fq(i)p Fm(=0)2446 815 y Fr(b)2482 827 y Fq(i)p Fm(+1)2594 815 y Fr(q)2631 827 y Fq(i)2677 815 y Fp(\000)2760 688 y Fq(n)2801 697 y Fi(k)2832 710 y(j)2868 688 y Fn(\000)p Fq(m)2809 736 y Fj(X)2815 913 y Fq(i)p Fm(=0)2992 815 y Fr(x)3039 827 y Fq(i)p Fm(+1)3151 815 y Fr(q)3188 827 y Fq(i)1151 1016 y Ft(=)e Fr(r)1335 1028 y Fq(n)1376 1037 y Fi(k)1407 1050 y(j)1444 1028 y Fn(\000)p Fq(m)1559 1016 y Ft(\()p Fr(b)p Ft(\))18 b Fp(\000)g Fr(r)1797 1028 y Fq(n)1838 1037 y Fi(k)1869 1050 y(j)1905 1028 y Fn(\000)p Fq(m)2020 1016 y Ft(\()p Fr(x)p Ft(\))24 b Fp(\024)f Fr(q)2280 1028 y Fq(n)2321 1037 y Fi(k)2352 1050 y(j)2388 1028 y Fn(\000)p Fq(m)p Fm(+1)2605 1016 y Ft(+)18 b Fr(q)2725 1028 y Fq(n)2766 1037 y Fi(k)2797 1050 y(j)2833 1028 y Fn(\000)p Fq(m)2966 1016 y Fp(\000)g Ft(1)g Fp(\000)g Fr(q)3229 1028 y Fq(n)3270 1037 y Fi(k)3301 1050 y(j)3337 1028 y Fn(\000)p Fq(m)1151 1145 y Fp(\024)82 b Fr(q)1335 1157 y Fq(n)1376 1166 y Fi(k)1407 1179 y(j)1443 1157 y Fn(\000)p Fq(m)p Fm(+1)1642 1145 y Fr(:)456 1296 y Ft(Since)27 b Fr(q)709 1308 y Fq(n)p Fm(+2)862 1296 y Ft(=)22 b Fr(a)993 1308 y Fq(n)p Fm(+2)1123 1296 y Fr(q)1160 1308 y Fq(n)p Fm(+1)1307 1296 y Ft(+)c Fr(q)1427 1308 y Fq(n)1496 1296 y Fp(\025)k Ft(2)p Fr(q)1662 1308 y Fq(n)1707 1296 y Ft(,)28 b(w)n(e)f(get)875 1458 y Fr(r)912 1470 y Fq(n)953 1479 y Fi(k)984 1492 y(j)1024 1458 y Ft(\()p Fr(b)p Ft(\))p 875 1514 250 4 v 925 1590 a Fr(q)962 1602 y Fq(n)1003 1611 y Fi(k)1034 1624 y(j)1153 1533 y Fp(\000)1245 1458 y Fr(r)1282 1470 y Fq(n)1323 1479 y Fi(k)1354 1492 y(j)1395 1458 y Ft(\()p Fr(x)p Ft(\))p 1245 1514 262 4 v 1302 1590 a Fr(q)1339 1602 y Fq(n)1380 1611 y Fi(k)1411 1624 y(j)1540 1533 y Fp(\024)1637 1458 y Fr(q)1674 1470 y Fq(n)1715 1479 y Fi(k)1746 1492 y(j)1782 1470 y Fn(\000)p Fq(m)p Fm(+1)p 1637 1514 344 4 v 1735 1590 a Fr(q)1772 1602 y Fq(n)1813 1611 y Fi(k)1844 1624 y(j)2014 1533 y Fp(\024)2112 1458 y Ft(2)2154 1422 y Fn(\000)2206 1428 y Fp(b)2252 1397 y Fi(m)p Ff(\000)p Fg(1)p 2252 1410 125 3 v 2300 1442 a(2)2386 1428 y Fp(c)2427 1458 y Fr(q)2464 1470 y Fq(n)2505 1479 y Fi(k)2536 1492 y(j)p 2112 1514 465 4 v 2270 1590 a Fr(q)2307 1602 y Fq(n)2348 1611 y Fi(k)2379 1624 y(j)2609 1533 y Ft(=)c(2)2739 1497 y Fn(\000)2791 1503 y Fp(b)2837 1472 y Fi(m)p Ff(\000)p Fg(1)p 2837 1485 125 3 v 2885 1517 a(2)2971 1503 y Fp(c)3012 1533 y Fr(:)456 1742 y Ft(and)691 1917 y(lim)14 b(sup)733 1987 y Fq(k)q Fn(!1)968 1860 y Fr(r)1005 1872 y Fq(k)1047 1860 y Ft(\()p Fr(x)p Ft(\))p 968 1898 191 4 v 1025 1974 a Fr(q)1062 1986 y Fq(k)1192 1917 y Fp(\025)22 b Ft(lim)14 b(sup)1325 1985 y Fq(j)s Fn(!1)1557 1855 y Fr(r)1594 1867 y Fq(k)1629 1875 y Fi(j)1665 1855 y Ft(\()p Fr(b)p Ft(\))p 1557 1898 209 4 v 1608 1974 a Fr(q)1645 1986 y Fq(k)1680 1994 y Fi(j)1794 1917 y Fp(\000)k Ft(2)1919 1881 y Fn(\000)1971 1887 y Fp(b)2017 1855 y Fi(m)p Ff(\000)p Fg(1)p 2017 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a Fr(q)1581 3026 y Fq(k)q Fm(+1)1724 3014 y Ft(+)k Fr(q)1844 3026 y Fq(k)1904 3014 y Fp(\000)g Ft(1)2038 2957 y Fr(;)99 b Fx(r)2199 2969 y Fm(1)2260 2957 y Ft(=)22 b(lim)14 b(sup)2390 3027 y Fq(k)q Fn(!1)2625 2901 y Fr(q)2662 2913 y Fq(k)q Fm(+1)p 2625 2938 162 4 v 2667 3014 a Fr(q)2704 3026 y Fq(k)456 3143 y Fh(Then)33 b Fx(r)714 3155 y Fm(0)780 3143 y Fp(\024)27 b Fr(R)p 872 3156 64 4 v 1 w Ft(\()p Fr(x)p Ft(\))i Fp(\024)p 1169 3076 V 28 w Fr(R)q Ft(\()p Fr(x)p Ft(\))g Fp(\024)e Fx(r)1504 3155 y Fm(1)1575 3143 y Fh(for)33 b(every)g Fr(x)c Fp(2)f Ft(\006)2151 3155 y Fq(\013)2231 3143 y Fh(and)33 b Fr(R)p 2395 3156 V 1 w Ft(\()p Fr(x)p Ft(\))c(=)f Fx(r)2731 3155 y Fm(0)2768 3143 y Fh(,)p 2827 3097 40 4 v 34 w Fr(r)s Ft(\()p Fr(x)p Ft(\))h(=)f Fx(r)3139 3155 y Fm(1)3209 3143 y Fh(almost)456 3243 y(everywher)l(e.)555 3342 y(If)40 b Fp(f)p Fr(a)738 3354 y Fq(k)778 3342 y Fp(g)820 3312 y Fn(1)820 3366 y Fq(k)q Fm(=0)984 3342 y Fh(is)g(unb)l(ounde)l(d,)i(then)e Fx(r)1762 3354 y Fm(0)1840 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b(S.,)f(priv)l(ate)h(comm)n(unication.)491 4967 y([3])35 b(Dartnell)16 b(P)-6 b(.,)18 b(Durand)f(F.,)g(Maass)f (A.:)27 b(Orbit)17 b(equiv)l(alence)h(and)g(Kakutani)g(equiv)l(alence)g (with)f(Sturmian)601 5050 y(subshifts.)23 b Fl(Studia)j(Math.)e Fu(142)p Fw(,)e(25-45)i(\(2000\))491 5133 y([4])35 b(Durand)g(F.,)j (Maass)d(A.:)54 b(Limit)33 b(la)n(ws)i(of)h(en)n(trance)h(times)d(for)h (lo)n(w-complexit)n(y)f(Can)n(tor)i(minimal)601 5216 y(systems.)22 b Fl(Nonline)l(arity)i Fu(14)p Fw(,)f(683-700)h(\(2001\)) p eop %%Page: 11 11 11 10 bop 1173 251 a Fm(LOCAL)29 b(RETURN)h(RA)-5 b(TES)28 b(IN)i(STURMIAN)f(SHIFTS)650 b(11)491 450 y Fw([5])35 b(Hardy)24 b(G.)h(H.,)f(W)-6 b(righ)n(t)25 b(E.)f(M.:)33 b(An)25 b(In)n(tro)r(duction)h(to)g(the)f(Theory)h(of)e(Num)n(b)r(ers.) f(Oxford)h(Univ)n(ersit)n(y)601 533 y(Press,)e(Oxford)i(1979.)491 616 y([6])35 b(Hedlund)20 b(G.,)g(Morse)g(M.:)28 b(Sym)n(b)r(olics)19 b(dynamics)g(I)r(I.)h(Sturmian)f(tra)t(jectories.)h Fl(A)n(m.)i(J.)h (Math.)d Fu(62)p Fw(,)f(1-42)601 699 y(\(1940\))491 782 y([7])35 b(Hirata)26 b(M.,)f(Saussol)h(B.,)g(V)-6 b(aien)n(ti)26 b(S.:)36 b(Statistics)27 b(of)f(return)g(times:)34 b(A)26 b(general)h(framew)n(ork)d(and)j(new)601 865 y(applications.)d Fl(Comm.)i(Math.)g(Phys.)e Fu(206)p Fw(,)e(3-55)i(\(1999\))491 948 y([8])35 b(K)-7 b(\027)-46 b(urk)l(a)23 b(P)-6 b(.:)31 b(Lo)r(cal)24 b(return)g(rates)g(in)f(substitutiv)n(e)i(subshifts,)e (submitted.)491 1031 y([9])35 b(Saussol)21 b(B.,)g(T)-6 b(roub)r(etzk)n(o)n(y)24 b(S.,)d(V)-6 b(aien)n(ti)22 b(S.:)29 b(Reccurence,)24 b(dimensions,)c(and)i(Ly)n(apuno)n(v)h(exp)r (onen)n(ts.)g Fl(J.)601 1114 y(Stat.)i(Phys.)f Fu(106)p Fw(,)e(3/4,)i(623-634)h(\(2002\))456 1197 y([10])35 b(S\023)-35 b(os)28 b(V.:)40 b(On)28 b(the)h(distribution)f(mo)r(d)g(1)g(of)g(the)h (sequences)h Fa(n\013)p Fw(.)e Fl(A)n(nn.)i(Univ.)f(Sci.)g(Budap)l(est) j(E\177)-36 b(otv\177)g(os)601 1280 y(Se)l(ct.)25 b(Math.)f Fu(1)p Fw(,)e(127-134)j(\(1958\))p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0303210633719--