%!PS (but not EPSF; comments have been disabled) %DVIPSCommandLine: dvips itm -o %DVIPSParameters: dpi=600, compressed, comments removed %DVIPSSource: TeX output 1998.02.19:1504 /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 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 0 0 0]N /nn 0 N /IE 0 N /ctr 0 N /df-tail{ /nn 8 dict N nn begin /FontType 3 N /FontMatrix fntrx N /FontBBox FBB N string /base X array /BitMaps X 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b(SCHMELING)h(AND)h(SER)n(GE)e(TR)n(OUBETZK)n(O)n(Y)754 1184 y Fu(Abstra)n(ct)p Fv(.)42 b(In)n(terv)-5 b(al)32 b(translation)f(mappings)h(are)f(not)h(necessarily)f(in-)754 1284 y(v)n(ertible)36 b(lo)r(cal)g(isometries)g(of)g(the)h(in)n(terv)-5 b(al)36 b(whic)n(h)h(where)f(recen)n(tly)g(in-)754 1383 y(tro)r(duced)31 b(b)n(y)f(Boshernitzan)g(and)g(Kornfeld)g([1].)46 b(Our)30 b(main)h(result)f(is)h(a)754 1483 y(sharp)24 b(upp)r(er)g(b)r(ound)g(on)g(the)g(n)n(um)n(b)r(er)g(of)g(minimal)g (sets)g(suc)n(h)f(a)h(mapping)754 1582 y(can)k(ha)n(v)n(e.)1546 2013 y Ft(1.)55 b Fs(Intr)n(oduction)555 2187 y Ft(An)44 b(imp)s(ortan)m(t)e(\014eld)h(in)g(the)h(study)g(of)f(dynamical)f (systems)j(are)f(the)g(in-)456 2303 y(v)m(estigations)h(on)h(v)m (olume{preserving)g(mappings.)83 b(These)48 b(mappings)d(ha)m(v)m(e)456 2420 y(strong)38 b(relations)f(to)h(Hamiltonian)d(systems.)62 b(A)39 b(ma)5 b(jor)37 b(feature)i(of)f(them)g(is)456 2536 y(that)k(pro)m(vided)h(these)h(maps)e(are)g(in)m(v)m(ertible)g 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Ft(Let)g(\001)2628 4178 y Fo(i)2688 4163 y Ft(=)c([)p Fr(\014)2877 4178 y Fo(i)p Fn(\000)p Fp(1)2996 4163 y Fr(;)17 b(\014)3095 4178 y Fo(i)3122 4163 y Ft(\))p Fr(:)35 b Ft(Then)456 4279 y(de\014ne)j Fr(T)14 b(x)37 b Ft(=)f Fr(x)25 b Ft(+)h Fr(\015)1250 4294 y Fo(i)1315 4279 y Ft(for)37 b Fr(x)f Fq(2)h Ft(\001)1744 4294 y Fo(i)1772 4279 y Fr(:)h Ft(The)g(only)f(restriction)g(on)g(the)h Fr(\015)3097 4294 y Fo(i)3163 4279 y Ft(is)f(that)456 4396 y Fr(T)14 b Ft(\001)608 4411 y Fo(i)663 4396 y Fq(\032)29 b Fr(I)8 b(:)32 b Ft(Let)h(\012\()p Fr(T)14 b Ft(\))28 b(=)f Fq(\\)1468 4359 y Fn(1)1468 4420 y Fo(n)p Fp(=0)1606 4396 y Fr(T)1677 4359 y Fo(n)1723 4396 y Fr(I)8 b(:)555 4512 y Ft(Similar)39 b(to)k(the)g(w)m(a)m(y)h(in)m(terv)-5 b(al)41 b(exc)m(hange)k(transformations)c(\(IETs\))j(arise)456 4628 y(in)34 b(p)s(olygonal)e(billiards)g(ITM's)k(arise)e(in)g(p)s (olygonal)f(billiards)e(with)j(semip)s(er-)456 4744 y(meable)39 b(w)m(alls)h(\(sp)m(y)h(mirrors\).)66 b(These)42 b(billiards)c(are)i (not)g(in)m(v)m(ertible)g(lo)s(cally)456 4861 y(measure)27 b(preserving)i(and)e(hence)i(globally)c(con)m(tract)j(area.)42 b(The)28 b(main)e(goal)g(of)456 4977 y(studying)k(suc)m(h)i(mo)s(dels)d (is)g(to)h(in)m(v)m(estigate)g(the)h(in)m(terpla)m(y)e(b)s(et)m(w)m (een)k(lo)s(cal)28 b(area)p 456 5122 499 4 v 555 5216 a Fm(Date)6 b Fv(:)28 b(F)-7 b(ebruary)27 b(10,)f(1997.)1931 5315 y Fl(1)p eop 2 1 bop 456 253 a Fl(2)524 b(J)1068 234 y(\177)1057 253 y(OR)n(G)33 b(SCHMELING)h(AND)e(SER)n(GE)h(TR)n(OUBETZK)n(O)n(Y)456 425 y Ft(preserv)-5 b(ation)32 b(and)h(global)d(loss)i(of)g(area.)44 b(This)32 b(mec)m(hanism)g(pro)m(vides)h(a)g(lot)e(of)456 541 y(in)m(teresting)h(prop)s(erties.)555 658 y(ITM's)j(where)g(recen)m (tly)g(in)m(tro)s(duced)f(b)m(y)h(Boshernitzan)f(and)f(Kornfeld)g([1].) 456 774 y(They)42 b(de\014ned)g(t)m(w)m(o)f(classes)h(of)e(ITM's:)61 b(those)42 b(of)e(\014nite)g(t)m(yp)s(e)i(and)e(those)i(of)456 890 y(in\014nite)25 b(t)m(yp)s(e.)42 b(They)28 b(sho)m(w)m(ed)g(that)e (is)g Fr(T)40 b Ft(is)26 b(of)g(\014nite)g(t)m(yp)s(e)h(then)g(\012\()p Fr(T)14 b Ft(\))27 b(consists)456 1006 y(of)34 b(a)g(\014nite)h(union)f (of)g(in)m(terv)-5 b(als)34 b(and)h Fr(T)14 b Fq(j)1974 1022 y Fp(\012\()p Fo(T)c Fp(\))2169 1006 y Ft(is)35 b(an)f(IET.)i(On)f(the)g(other)g(hand)456 1123 y(they)f(constructed)i (examples)e(of)f(ITM's)i(of)f(in\014nite)f(t)m(yp)s(e)h(for)g(whic)m(h) g(\012\()p Fr(T)14 b Ft(\))34 b(is)456 1239 y(a)e(Can)m(tor)h(set)g(of) f(fractional)e(dimension.)555 1355 y(Our)38 b(main)d(result,)k(theorem) e(2.4)g(is)g(an)g(upp)s(er)h(b)s(ound)g(on)f(the)h(n)m(um)m(b)s(er)g (of)456 1471 y(minimal)g(sets)44 b(a)e(ITM)h(can)g(ha)m(v)m(e.)75 b(In)42 b(theorem)h(2.10)f(w)m(e)h(sho)m(w)h(this)e(upp)s(er)456 1588 y(b)s(ound)c(is)h(sharp.)62 b(W)-8 b(e)39 b(also)f(start)h(the)g (study)h(of)e(t)m(ypicalit)m(y)g(of)g(\014nite)g(v)m(ersus)456 1704 y(in\014nite)31 b(t)m(yp)s(e)j(\(theorems)e(2.5-2.8\).)982 1984 y(2.)55 b Fs(Not)-7 b(a)g(tion)40 b(and)e(st)-7 b(a)g(tement)37 b(of)h(resul)-7 b(ts)555 2159 y Ft(W)f(e)36 b(mak)m(e)f(the)h(con)m(v)m(en)m(tion)h(that)e(the)g(w)m(ord)h(in)m (terv)-5 b(al)34 b(will)f(alw)m(a)m(ys)j(refer)g(to)456 2275 y(an)c(in)m(terv)-5 b(al)31 b(whic)m(h)i(is)g(closed)f(on)h(the)g (left,)f(op)s(en)g(on)h(the)g(righ)m(t.)555 2391 y(Let)41 b(\012)808 2406 y Fp(0)890 2391 y Ft(=)g Fr(I)49 b Ft(and)41 b(\012)1367 2406 y Fo(n)1456 2391 y Ft(=)g Fr(T)1644 2355 y Fo(n)1691 2391 y Ft(\012)1761 2406 y Fp(0)1801 2391 y Fr(:)g Ft(Eac)m(h)h(\012)2190 2406 y Fo(n)2278 2391 y Ft(is)e(a)h(\014nite)f(union)g(of)h(disjoin)m(t)456 2507 y(in)m(terv)-5 b(als)32 b(\012)917 2471 y Fo(i)917 2532 y(n)998 2507 y Ft(and)h(\012)1258 2522 y Fo(n)1334 2507 y Fq(\033)c Ft(\012)1510 2522 y Fo(n)p Fp(+1)1681 2507 y Ft(for)k(all)e Fr(n:)i Ft(Let)h(\012)29 b(=)f Fq(\\)2530 2522 y Fo(n)p Fn(\025)p Fp(0)2668 2507 y Ft(\012)2738 2522 y Fo(n)2818 2507 y Ft(and)34 b Fr(X)i Ft(=)p 3231 2427 71 4 v 29 w(\012.)46 b Fr(T)456 2624 y Ft(is)36 b(called)g(an)h(ITM)h(of)e(\014nite)h(t)m(yp)s(e)h(if)d(\012)1967 2639 y Fo(n)2050 2624 y Ft(=)g(\012)2231 2639 y Fo(n)p Fp(+1)2406 2624 y Ft(for)h(some)h Fr(n)p Ft(.)57 b(In)37 b(this)f(case)456 2740 y(\012)28 b(=)f(\012)727 2755 y Fo(n)775 2740 y Fr(:)32 b Ft(If)h Fr(T)46 b Ft(is)32 b(not)h(of)f(\014nite)g(t)m(yp)s(e)h(w)m(e)h(call)d(it)g(of)i (in\014nite)e(t)m(yp)s(e.)555 2856 y(If)c Fr(T)40 b Ft(is)26 b(of)g(\014nite)g(t)m(yp)s(e,)i(then)f Fr(X)35 b Ft(consists)27 b(of)f(a)g(\014nite)g(union)g(of)g(in)m(terv)-5 b(als)25 b(and)456 2972 y(the)k(restriction)f(of)h Fr(T)43 b Ft(to)28 b Fr(X)37 b Ft(is)29 b(an)g(in)m(terv)-5 b(al)28 b(exc)m(hange)j (transformation)3148 2936 y Fp(1)3185 2972 y Ft(.)42 b(This)456 3089 y(w)m(as)31 b(already)e(observ)m(ed)k(in)c([1].)43 b(Our)30 b(\014rst)g(theorem)g(states)h(that)f(the)h(con)m(v)m(erse)456 3205 y(is)h(also)f(true.)456 3396 y Fw(Theorem)37 b(2.1.)49 b Fk(If)41 b Fr(X)49 b Fk(c)-5 b(onsists)41 b(of)g(a)g(\014nite)g 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Fr(;)g Fq(\001)g(\001)g(\001)31 b Fr(;)17 b(\015)2816 4067 y Fo(r)2853 4052 y Ft(\))29 b(whic)m(h)g(de\014ne)456 4168 y(an)f(ITM.)i(In)f([1])g(it)e(is)i(pro)m(v)m(ed)h(that)e(an)m(y)i(ITM)f (of)f(rank)i(2)e(is)g(of)g(\014nite)h(t)m(yp)s(e.)43 b(W)-8 b(e)456 4284 y(ha)m(v)m(e)34 b(the)f(follo)m(wing)c(statemen)m (t.)456 4475 y Fw(Corollary)35 b(2.2.)50 b Fk(A)n(ny)34 b(ITM)h(with)g(two)g(br)-5 b(anches)33 b(only)i(is)g(of)f(\014nite)h (typ)-5 b(e.)456 4666 y Fw(Remark:)50 b Ft(The)37 b(maximal)32 b(rank)37 b(whic)m(h)f(an)g(ITM's)i(with)e(t)m(w)m(o)g(branc)m(hes)i (can)456 4782 y(ha)m(v)m(e)33 b(is)f(4.)43 b(Th)m(us)34 b(the)f(previous)g(corollary)e(is)g(not)i(con)m(tained)f(in)g(the)h (theorem)p 456 4925 499 4 v 555 4986 a Fi(1)592 5016 y Fv(W)-7 b(e)34 b(broaden)f(the)g(de\014nition)h(of)g(ITMs)f(and)g (IETs)g(here,)i(w)n(e)e(call)g Fh(T)44 b Fv(an)33 b(ITM)h(if)g Fh(T)45 b Fv(is)456 5116 y(not)25 b(de\014ned)i(on)e(some)g(of)h(the)g (in)n(terv)-5 b(als)25 b(\001)1839 5128 y Fg(i)1893 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Fq([)2529 1075 y Fo(r)2529 1136 y(i)p Fp(=1)2669 1111 y Fq([)2735 1070 y Fp(+)p Fn(1)2735 1133 y Fo(n)p Fp(=)p Fn(\0001)2985 1111 y Fr(T)3056 1075 y Fo(n)3103 1111 y Fr(\014)3158 1126 y Fo(i)3208 1111 y Fq([)22 b(f)p Ft(1)p Fq(g)456 1227 y Ft(and)27 b Fr(B)33 b Ft(:=)27 b Fr(I)19 b Fq([)11 b Fr(A)27 b Ft(\(disjoin)m(t)f(union\).)41 b(W)-8 b(e)27 b(in)m(tro)s(duce)g(an)g(order)g(on)g Fr(B)32 b Ft(b)m(y)c(taking)456 1344 y(the)40 b(natural)g(linear)e(orders)j(on) f(b)s(oth)h Fr(A)f Ft(and)g Fr(I)8 b Ft(,)42 b(then)f(putting)f(ev)m (ery)i(p)s(oin)m(t)456 1460 y Fr(x)28 b Fq(2)g Fr(A)k Ft(directly)f(to)h(the)h(left)e(of)h(its)f(coun)m(terpart)i(in)e Fr(I)40 b Ft(and)32 b(putting)g(the)g(p)s(oin)m(t)456 1576 y(1)27 b Fq(2)h Fr(A)j Ft(at)f(the)h(far)f(righ)m(t.)42 b(It)30 b(is)g(easy)i(to)e(c)m(hec)m(k)j(that)d(the)h(corresp)s(onding) f(order)456 1692 y(top)s(ology)g(mak)m(es)j Fr(B)k Ft(in)m(to)32 b(a)f(compact,)h(metrizable)f(space,)i(and)f(that)g Fr(T)46 b Ft(has)33 b(a)456 1808 y(unique)g(con)m(tin)m(uous)g(extension)g(to)g Fr(B)k Ft([2],[1].)555 1925 y(W)-8 b(e)34 b(sa)m(y)g Fr(T)47 b Ft(is)33 b(ap)s(erio)s(dic)e(if)h Fr(O)1687 1889 y Fp(+)1745 1925 y Ft(\()p Fr(x)p Ft(\))i(is)e(in\014nite)h(for)f (all)f Fr(x)e Fq(2)h Fr(I)8 b Ft(.)45 b(W)-8 b(e)33 b(sa)m(y)i(that)456 2041 y Fr(Y)51 b Ft(is)29 b(a)g(minimal)d(set)k(if)f Fr(Y)51 b Ft(is)29 b(a)g(closed)h(subset)i(\(of)d Fr(B)5 b Ft(\),)2503 2005 y Fp(2)2573 2041 y Fr(T)14 b(Y)48 b Fq(\032)28 b Fr(Y)22 b Ft(,)30 b(and)g Fr(O)3255 2005 y Fp(+)3313 2041 y Ft(\()p Fr(x)p Ft(\))456 2157 y(is)j(dense)i(in)d Fr(Y)55 b Ft(for)33 b(all)e Fr(x)f Fq(2)f Fr(Y)5 b(:)34 b Ft(Our)f(main)f(result)i(is)f(an)g(upp)s(er)h(b)s(ound)g(on)f(the)456 2273 y(n)m(um)m(b)s(er)g(of)f(minimal)c(sets)34 b(an)e(ITM)i(can)f(ha)m (v)m(e.)456 2442 y Fw(Theorem)k(2.4.)49 b Fk(Supp)-5 b(ose)39 b Fr(T)54 b Fk(is)40 b(ap)-5 b(erio)g(dic.)59 b(The)40 b(numb)-5 b(er)40 b(of)g(minimal)e(sets)456 2558 y(is)c(less)h(than)f(or)h(e)-5 b(qual)35 b(to)g Ft([)p Fr(r)s(=)p Ft(2])p Fr(:)555 2727 y Ft(The)f(next)f(statemen)m (ts)h(are)e(concerning)h(the)g(t)m(ypical)f(b)s(eha)m(vior)g(of)g (ITM's.)456 2895 y Fw(Theorem)37 b(2.5.)49 b Fk(The)34 b(set)h(of)g(ITM's)f(of)g(in\014nite)h(typ)-5 b(e)35 b(is)f(a)h Fr(G)2840 2910 y Fo(\016)2878 2895 y Fk(.)555 3064 y Ft(This)e(theorem)f(has)h(t)m(w)m(o)h(corollaries:)456 3233 y Fw(Corollary)h(2.6.)50 b Fk(The)33 b(set)h(of)g(ITMs)g(for)g (which)f Fr(X)42 b Fk(has)34 b(zer)-5 b(o)34 b(L)-5 b(eb)g(esgue)33 b(me)-5 b(a-)456 3349 y(sur)g(e)35 b(is)f(a)h Fr(G)930 3364 y Fo(\016)968 3349 y Fk(.)456 3517 y Fw(Corollary)g(2.7.)50 b Fk(If)d(ITM's)g(of)h(in\014nite)g(typ)-5 b(e)48 b(ar)-5 b(e)48 b(somewher)-5 b(e)46 b(dense)i(then)456 3634 y(they)35 b(form)f(a)h(set)g(of)f(se)-5 b(c)g(ond)34 b(Bair)-5 b(e)34 b(c)-5 b(ate)g(gory.)555 3802 y Ft(In)33 b(con)m(trast)g(w)m(e)h (ha)m(v)m(e)g(the)f(follo)m(wing)d(results.)456 3971 y Fw(Theorem)37 b(2.8.)49 b Fk(F)-7 b(or)37 b(any)h Fr(r)e(>)e Ft(1)k Fk(ther)-5 b(e)38 b(is)f(an)h(op)-5 b(en)38 b(set)g(of)f(ITM's)h (of)g(\014nite)456 4087 y(typ)-5 b(e.)456 4256 y Fw(Theorem)37 b(2.9.)49 b Fk(If)f Fr(T)63 b Fk(is)49 b(a)h(maximal)d(r)-5 b(ank)49 b(ITM)g(of)g(\014nite)g(typ)-5 b(e)50 b(then)f(al)5 b(l)456 4372 y(ITM's)34 b(in)g(a)h(neighb)-5 b(orho)g(o)g(d)33 b(of)i Fr(T)48 b Fk(ar)-5 b(e)35 b(of)g(\014nite)f(typ)-5 b(e.)555 4541 y Ft(Keane)38 b(has)h(sho)m(wn)g(that)e(a)h(generic)g (IET)g(is)g(minimal)33 b([2].)59 b(The)39 b(situation)456 4657 y(c)m(hanges)34 b(drastically)c(in)i(the)h(case)h(of)e(ITM's.)456 4825 y Fw(Theorem)37 b(2.10.)49 b Fk(F)-7 b(or)35 b(e)-5 b(ach)35 b Fr(r)e(>)d Ft(1)35 b Fk(ther)-5 b(e)36 b(is)g(an)g(op)-5 b(en)35 b(set)h(of)g(ITM's)f(having)456 4942 y(at)g(le)-5 b(ast)34 b Ft([)p Fr(r)s(=)p Ft(2])h Fk(minimal)e(c)-5 b(omp)g(onents.)p 456 5025 499 4 v 555 5086 a Fi(2)592 5116 y Fv(Alternativ)n(ely)e(,)32 b(if)g Fh(Y)49 b Ff(\032)29 b Fh(I)38 b Fv(then)32 b Fh(Y)51 b Fv(should)31 b(b)r(e)h(closed)f (from)g(the)h(left,)h(and)e(op)r(en)h(from)456 5216 y(the)c(righ)n(t.)p eop 4 3 bop 456 253 a Fl(4)524 b(J)1068 234 y(\177)1057 253 y(OR)n(G)33 b(SCHMELING)h(AND)e(SER)n(GE)h(TR)n(OUBETZK)n(O)n(Y)1489 425 y Ft(3.)55 b Fs(Eventual)37 b(IET's)456 600 y Fw(Pro)s(of)k(of)h (theorem)f(2.1:)51 b Ft(F)-8 b(or)35 b Fr(x)i Ft(prep)s(erio)s(dic)e (de\014ne)i(its)f(prep)s(erio)s(d)f(to)h(b)s(e)456 716 y Fr(k)29 b Ft(+)d Fr(l)41 b Ft(where)f Fr(T)1067 680 y Fo(i)1095 716 y Fr(x)e Ft(is)h(not)f(p)s(erio)s(dic)f(for)h(0)g Fq(\024)g Ft(1)g Fr(<)g(k)s Ft(,)i(while)e Fr(T)2863 680 y Fo(k)2905 716 y Fr(x)h Ft(is)f(p)s(erio)s(dic)456 832 y(with)32 b(p)s(erio)s(d)f Fr(l)r(:)i Ft(Supp)s(ose)h Fr(X)40 b Ft(consist)33 b(of)f Fr(N)43 b Ft(disjoin)m(t)31 b(in)m(terv)-5 b(als.)43 b(W)-8 b(e)33 b(set)842 1092 y Fr(\017)28 b Ft(:=)1050 1025 y(1)p 1050 1069 49 4 v 1050 1160 a(2)1125 1092 y(min)1304 922 y Fe(\()1440 967 y Fo(r)1384 997 y Fe(X)1392 1209 y Fo(k)r Fp(=0)1545 1092 y Fr(n)1603 1107 y Fo(k)1646 1092 y Fr(\015)1697 1107 y Fo(k)1767 1092 y Ft(:)93 b Fq(k)p Fr(n)1995 1107 y Fo(k)2037 1092 y Fq(k)28 b Fr(<)f Ft(2)p Fr(N)10 b Ft(;)2454 967 y Fo(r)2399 997 y Fe(X)2407 1209 y Fo(k)r Fp(=0)2560 1092 y Fr(n)2618 1107 y Fo(k)2661 1092 y Fr(\015)2712 1107 y Fo(k)2782 1092 y Fq(6)p Ft(=)27 b(0)2934 922 y Fe(\))3031 1092 y Fr(:)456 1366 y Ft(Since)39 b Fr(T)788 1330 y Fo(j)824 1366 y Fr(x)h Ft(has)f(the)h(form)1510 1291 y Fe(P)1616 1317 y Fo(r)1616 1395 y(k)r Fp(=0)1765 1366 y Fr(n)1823 1381 y Fo(k)1866 1366 y Fr(\015)1917 1381 y Fo(k)1986 1366 y Ft(+)26 b Fr(x)40 b Ft(with)e Fq(k)p Fr(n)2519 1381 y Fo(k)2562 1366 y Fq(k)h(\024)g Fr(j)45 b Ft(w)m(e)40 b(ha)m(v)m(e)g(for)f Fr(x)456 1482 y Ft(ap)s(erio)s(dic,)g(p)s(erio)s(dic)f(with)h(p)s(erio)s(d)g(greater) g(than)h(2)p Fr(N)10 b Ft(,)41 b(or)e(prep)s(erio)s(dic)g(with)456 1598 y(prep)s(erio)s(d)d(greater)h(than)g(2)p Fr(N)48 b Ft(that)36 b Fr(d)p Ft(\()p Fr(T)2016 1562 y Fo(i)2044 1598 y Fr(x;)17 b(T)2214 1562 y Fo(j)2250 1598 y Fr(x)p Ft(\))36 b Fr(>)f Ft(2)p Fr(\017)i Ft(for)g(all)e(0)g Fq(\024)h Fr(i)f(<)h(j)41 b Fq(\024)456 1714 y Ft(2)p Fr(N)10 b Ft(.)555 1831 y(W)-8 b(e)31 b(consider)f(an)g Fr(\017)p Ft({neigh)m(b)s(orho)s(o)s(d)f Fr(U)41 b Ft(of)30 b Fr(X)38 b Ft(consisting)29 b(of)h Fr(X)38 b Ft(and)30 b(a)g(union)456 1947 y Fq([)522 1911 y Fp(2)p Fo(N)522 1973 y(k)r Fp(=1)655 1947 y Fr(I)698 1962 y Fo(k)774 1947 y Ft(where)35 b Fr(I)1100 1962 y Fo(k)1176 1947 y Ft(are)e(in)m(terv)-5 b(als)33 b(of)g(length)g(at)g(most)f Fr(\017)p Ft(.)47 b(T)-8 b(ak)m(e)34 b Fr(M)45 b Ft(su\016cien)m(tly) 456 2063 y(large)31 b(that)h(\012)975 2078 y Fo(M)1083 2063 y Fq(2)c Fr(U)10 b Ft(.)555 2179 y(If)31 b Fr(x)g Ft(is)f(p)s(erio)s(dic)f(then)j(there)f(is)f(an)h(in)m(terv)-5 b(al)29 b Fr(J)40 b Ft(ab)s(out)30 b Fr(x)h Ft(suc)m(h)h(that)f Fr(J)37 b Fq(\032)28 b Fr(X)r(:)456 2296 y Ft(If)36 b Fr(x)i Ft(is)e(prep)s(erio)s(dic)g(with)g(p)s(erio)s(d)g(less)h(than)g (or)g(equal)f(to)h(2)p Fr(N)47 b Ft(then)37 b(there)h(is)456 2412 y(an)32 b(in)m(terv)-5 b(al)32 b Fr(J)42 b Ft(ab)s(out)32 b Fr(x)h Ft(suc)m(h)i(that)d Fr(T)1910 2376 y Fp(2)p Fo(N)2013 2412 y Fr(J)37 b Fq(\032)28 b Fr(X)41 b Ft(and)33 b(th)m(us)h Fr(T)2807 2376 y Fp(2)p Fo(N)7 b Fp(+)p Fo(M)3039 2412 y Fr(J)37 b Fq(\032)29 b Fr(X)40 b Ft(as)456 2528 y(w)m(ell.)555 2644 y(Finally)33 b(supp)s(ose)k Fr(x)f Ft(is)f(ap)s(erio)s(dic,)f(or)h(has)h(\(pre\)p)s(erio)s(d)f(greater)g (than)g(2)p Fr(N)5 b(:)456 2761 y Ft(Then)26 b Fr(T)774 2724 y Fo(M)853 2761 y Fr(x)i Fq(2)g Fr(U)10 b Ft(.)42 b(If)25 b Fr(T)1336 2724 y Fo(M)1415 2761 y Fr(x)j Fq(2)g Fr(X)33 b Ft(w)m(e)26 b(are)g(done.)41 b(So)25 b(w)m(e)i(assume)f Fr(T)2932 2724 y Fo(M)3010 2761 y Fr(x)i Fq(2)g Fr(U)18 b Fq(n)7 b Fr(X)h Ft(.)456 2877 y(W)-8 b(e)47 b(claim)d(that)i Fr(T)1208 2841 y Fo(M)7 b Fp(+2)p Fo(N)1440 2877 y Fr(x)52 b Fq(2)g Fr(X)8 b Ft(.)86 b(F)-8 b(or)46 b(otherwise,)k Fr(T)2605 2841 y Fo(M)7 b Fp(+)p Fo(k)2778 2877 y Fr(x)52 b Fq(2)g Fr(U)42 b Fq(n)32 b Fr(X)54 b Ft(for)456 2993 y(0)40 b Fq(\024)i Fr(k)i Fq(\024)d Ft(2)p Fr(N)51 b Ft(and)40 b Fr(d)p Ft(\()p Fr(T)1412 2957 y Fo(M)7 b Fp(+)p Fo(j)1578 2993 y Fr(x;)17 b(T)1748 2957 y Fo(M)7 b Fp(+)p Fo(i)1905 2993 y Fr(x)p Ft(\))42 b Fr(>)e Ft(2)p Fr(\017)h Ft(for)f(all)e(0)j Fq(\024)g Fr(i)g(<)g(j)47 b Fq(\024)41 b Ft(2)p Fr(N)10 b Ft(,)43 b(a)456 3109 y(con)m(tradiction.)555 3225 y(W)-8 b(e)33 b(ha)m(v)m(e)h(sho)m(wn)g (\012)1314 3240 y Fo(M)7 b Fp(+2)p Fo(N)1574 3225 y Fq(\032)29 b Fr(X)40 b Ft(and)33 b(th)m(us)g Fr(T)46 b Ft(is)32 b(of)h(\014nite)f(t)m(yp)s(e.)p Fw(q.e.d.)456 3400 y(Pro)s(of)40 b(of)g(corollary)e(2.2:)49 b Ft(Since)35 b Fr(T)49 b Ft(has)35 b(only)f(t)m(w)m(o)i(branc)m(hes)h(and)e Fr(T)14 b(I)39 b Fq(\032)33 b Fr(I)456 3516 y Ft(\012)526 3531 y Fp(1)593 3516 y Ft(=)28 b Fr(T)14 b(I)37 b Ft(is)29 b(a)g(single)f(in)m(terv)-5 b(al.)41 b(By)30 b(induction)e(\012)2328 3531 y Fo(n)2405 3516 y Ft(is)h(a)g(single)f(in)m(terv)-5 b(al)29 b(for)f(all)456 3632 y Fr(n)p Ft(.)41 b(Hence,)29 b Fr(X)34 b Ft(is)25 b(either)h(an)g(in)m(terv)-5 b(al)24 b(or)i(a)g(p)s(oin)m(t.)40 b(The)27 b(latter)e(case)i(con)m(tradicts) 456 3748 y(ap)s(erio)s(dicit)m(y)-8 b(.)40 b(In)30 b(the)g(\014rst)h (case)f Fr(T)44 b Ft(is)29 b(of)h(\014nite)f(t)m(yp)s(e)i(b)m(y)g (theorem)e(2.1.)42 b Fw(q.e.d.)1317 3995 y Ft(4.)55 b Fs(ITM's)37 b(of)h(infinite)g(type)456 4169 y Fw(Pro)s(of)g(of)h (theorem)f(2.3:)46 b Ft(First)33 b(of)g(all)e Fr(X)42 b Ft(can)34 b(not)f(b)s(e)h(a)g(\014nite)f(set.)48 b(If)33 b(it)g(is)456 4286 y(\014nite)28 b(then)h(there)g(is)f(an)g Fr(n)g Fq(\025)g Ft(1)g(so)g(that)h Fr(T)2051 4250 y Fo(n)2126 4286 y Ft(\014xes)g(a)f(p)s(oin)m(t.)42 b(Then)29 b(it)e(is)h(easy)i(to)456 4402 y(see)37 b(that)g Fr(T)904 4366 y Fo(n)987 4402 y Ft(\014xes)h(an)e(in)m(terv)-5 b(al)36 b(of)g(p)s(oin)m(ts.)55 b(This)36 b(con)m(tradicts)h(the)g (fact)g(that)456 4518 y Fr(X)j Ft(is)32 b(\014nite.)555 4634 y(W)-8 b(e)34 b(next)g(sho)m(w)g(that)f Fr(X)41 b Ft(can)33 b(not)g(con)m(tain)g(an)g(in)m(terv)-5 b(al.)44 b(The)34 b(image)e(of)g(an)456 4751 y(in)m(terv)-5 b(al)30 b(is)i(a)f(\014nite)h(union)f(of)h(disjoin)m(t)f(in)m(terv)-5 b(als,)31 b(th)m(us)i(if)e(it)g(do)s(es)h(con)m(tain)f(an)456 4867 y(in)m(terv)-5 b(al)36 b(then)i Fr(X)46 b Ft(con)m(tains)37 b(a)h(union)f(of)g(disjoin)m(t)f(in)m(terv)-5 b(als)37 b Fr(X)44 b Fq(\033)37 b([)p Fr(I)3129 4882 y Fo(i)3158 4867 y Fr(:)g Ft(Since)456 4983 y Fr(T)14 b Fq(j)555 4998 y Fo(X)659 4983 y Ft(is)38 b(top)s(ologically)c(transitiv)m(e)j(w) m(e)i(ha)m(v)m(e)h Fr(X)k Ft(=)37 b Fq([)p Fr(I)2509 4998 y Fo(i)2538 4983 y Fr(:)h Ft(By)g(theorem)g(2.1)g(the)456 5099 y(union)46 b(can)h(not)g(b)s(e)g(a)f(\014nite)h(one)g(since)g (then)h Fr(T)60 b Ft(is)47 b(of)f(\014nite)h(t)m(yp)s(e.)87 b(Th)m(us)456 5216 y(it)40 b(is)g(a)h(coun)m(table)g(union.)68 b(W)-8 b(e)42 b(\(partially\))c(order)j(the)h(in)m(terv)-5 b(als)40 b(b)m(y)i(length)p eop 5 4 bop 1224 236 a Fl(INTER)-9 b(V)g(AL)33 b(TRANSLA)-6 b(TION)30 b(MAPPINGS)730 b(5)456 425 y Ft(and)41 b(assume)g(the)g(en)m (umeration)g(of)f(the)i Fr(I)2084 440 y Fo(i)2153 425 y Ft(is)e(compatible)f(with)i(this)f(order,)456 541 y(i.e.)57 b(if)37 b(the)g(length)g Fq(j)p Fr(I)1279 556 y Fo(i)1307 541 y Fq(j)g Ft(of)g Fr(I)1531 556 y Fo(i)1597 541 y Ft(is)g(strictly)g(greater)g(than)h(the)f(length)g Fq(j)p Fr(I)3153 556 y Fo(j)3190 541 y Fq(j)g Ft(then)456 658 y Fr(i)46 b(<)g(j:)d Ft(If)h(the)f(in)m(terv)-5 b(al)42 b Fr(I)1460 673 y Fo(i)1532 658 y Ft(do)s(es)h(not)g(con)m(tain)g(an)m (y)h(discon)m(tin)m(uit)m(y)f(p)s(oin)m(t)g Fr(\014)3408 673 y Fo(j)456 774 y Ft(the)k Fr(f)11 b Fq(j)725 789 y Fo(I)756 799 y Fd(i)832 774 y Ft(is)47 b(con)m(tin)m(uous)g(and)g(th) m(us)h Fr(f)58 b Ft(maps)46 b Fr(I)2298 789 y Fo(i)2373 774 y Ft(in)m(to)g(one)h(of)g(the)g(in)m(terv)-5 b(als)456 890 y Fr(I)499 905 y Fo(j)573 890 y Ft(whic)m(h)39 b(is)f(not)g (shorter)g(than)g Fr(I)1753 905 y Fo(i)1782 890 y Fr(:)g Ft(Only)g(a)f(\014nite)h(n)m(um)m(b)s(er)h(of)f(the)g(in)m(terv)-5 b(als)456 1006 y(con)m(tain)25 b(a)h(discon)m(tin)m(uit)m(y)g(p)s(oin)m (t.)40 b(Their)26 b(images)f(can)h(b)s(e)g(con)m(tained)h(in)e(shorter) 456 1123 y(in)m(terv)-5 b(als.)42 b(Since)32 b(there)h(are)f(only)g (\014nitely)f(man)m(y)h(discon)m(tin)m(uit)m(y)g(p)s(oin)m(ts)g(there) 456 1239 y(is)g(a)h(shortest)h(in)m(terv)-5 b(al)32 b Fr(I)1402 1254 y Fo(i)1426 1263 y Fc(0)1498 1239 y Ft(whic)m(h)h(con)m (tains)g(the)h(image)d(of)i(a)f(longer)h(in)m(terv)-5 b(al.)456 1355 y(There)24 b(is)f(also)g(a)g(shortest)h(in)m(terv)-5 b(al)22 b Fr(I)1822 1370 y Fo(i)1846 1379 y Fc(1)1908 1355 y Ft(whic)m(h)i(con)m(tains)g(a)f(discon)m(tin)m(uit)m(y)g(p)s (oin)m(t.)456 1471 y(F)-8 b(urthermore,)32 b(if)f Fr(I)1180 1486 y Fo(i)1241 1471 y Ft(and)h Fr(I)1473 1486 y Fo(j)1542 1471 y Ft(are)h(shorter)g(than)g(min)n(\()p Fq(j)p Fr(I)2535 1486 y Fo(i)2559 1495 y Fc(0)2598 1471 y Fq(j)p Fr(;)17 b Fq(j)p Fr(I)2741 1486 y Fo(i)2765 1495 y Fc(1)2802 1471 y Fq(j)p Ft(\))32 b(and)h Fr(T)14 b(I)3204 1486 y Fo(i)3260 1471 y Fq(\032)28 b Fr(I)3408 1486 y Fo(j)456 1588 y Ft(then)35 b Fr(T)14 b(I)794 1603 y Fo(i)854 1588 y Ft(=)32 b Fr(I)1005 1603 y Fo(j)1041 1588 y Fr(:)j Ft(Th)m(us)i(in\014nitely)d(man)m(y)h(of)f(the)i Fr(I)2360 1603 y Fo(i)2423 1588 y Ft(ha)m(v)m(e)g(the)g(same)e(length,)i(a)456 1704 y(con)m(tradiction.)555 1820 y(Finally)k(w)m(e)j(sho)m(w)g(that)f Fr(X)50 b Ft(can)43 b(not)f(ha)m(v)m(e)i(an)e(isolated)f(p)s(oin)m(t.) 71 b(Supp)s(ose)456 1936 y(the)42 b(p)s(oin)m(t)f Fr(x)k Fq(2)f Fr(X)50 b Ft(is)42 b(isolated.)70 b(Then)44 b(eac)m(h)f Fr(y)k Fq(2)d Fr(O)2547 1900 y Fp(+)2605 1936 y Ft(\()p Fr(x)p Ft(\))f(is)e(also)h(isolated.)456 2052 y(If)36 b Fr(O)635 2016 y Fp(+)693 2052 y Ft(\()p Fr(x)p Ft(\))h(is)f(\014nite) g(w)m(e)i(ha)m(v)m(e)g(a)e(con)m(tradiction)f(as)i(b)s(efore.)55 b(Th)m(us)38 b(w)m(e)g(assume)456 2169 y Fr(O)534 2133 y Fp(+)592 2169 y Ft(\()p Fr(x)p Ft(\))c(is)g(coun)m(table.)48 b(Since)34 b Fr(y)f Fq(2)e Fr(O)1853 2133 y Fp(+)1911 2169 y Ft(\()p Fr(x)p Ft(\))j(is)g(isolated)f(w)m(e)i(can)f(asso)s (ciate)g(to)f Fr(y)456 2285 y Ft(an)27 b(in)m(terv)-5 b(al)25 b Fr(I)975 2300 y Fo(y)1044 2285 y Ft(as)i(follo)m(ws.)41 b(Denote)27 b(b)m(y)h(\012)2044 2249 y Fo(i)2044 2310 y(n)2118 2285 y Ft(the)g(in)m(terv)-5 b(als)26 b(whic)m(h)i(compromise) 456 2401 y(\012)526 2416 y Fo(n)573 2401 y Fr(:)34 b Ft(The)h(in)m(terv)-5 b(al)33 b Fr(I)1233 2416 y Fo(y)1309 2401 y Ft(is)g(the)i(maximal)c(in)m(terv)-5 b(al)33 b(\012)2402 2365 y Fo(i)2402 2426 y(n)2483 2401 y Ft(con)m(taining)g Fr(y)k Ft(suc)m(h)f(that)456 2517 y Fr(I)499 2532 y Fo(y)562 2517 y Fq(\\)23 b Fr(X)35 b Ft(=)28 b Fr(y)t(:)k Ft(By)h(de\014nition)e (the)i(in)m(terv)-5 b(als)32 b Fr(I)2170 2532 y Fo(y)2244 2517 y Ft(are)g(all)f(disjoin)m(t.)555 2634 y(If)48 b(the)g(in)m(terv) -5 b(al)47 b Fr(I)1262 2649 y Fo(y)1351 2634 y Ft(do)s(es)h(not)g(con)m (tain)f(a)g(discon)m(tin)m(uit)m(y)h(p)s(oin)m(t)f(then)h(w)m(e)456 2750 y(claim)39 b(that)i Fr(T)14 b(I)1059 2765 y Fo(y)1143 2750 y Ft(=)42 b Fr(I)1304 2765 y Fo(T)10 b(y)1397 2750 y Fr(:)41 b Ft(Since)h Fr(I)1772 2765 y Fo(y)1856 2750 y Ft(=)h(\012)2045 2714 y Fo(i)2045 2774 y(n)2133 2750 y Ft(and)f Fr(T)14 b(I)2446 2765 y Fo(y)2528 2750 y Ft(is)41 b(connected)j(w)m(e)e(ha)m(v)m(e)456 2877 y Fr(T)14 b(I)570 2892 y Fo(y)638 2877 y Ft(=)28 b(\012)812 2830 y Fo(j)812 2902 y(n)p Fp(+1)973 2877 y Ft(for)c(some)f Fr(j)34 b Ft(=)27 b Fr(j)6 b Ft(\()p Fr(i;)17 b(n)p Ft(\))p Fr(:)24 b Ft(Clearly)f Fr(T)14 b(y)31 b Fq(2)d Ft(\012)2478 2830 y Fo(j)2478 2902 y(n)p Fp(+1)2639 2877 y Ft(and)c(\012)2890 2830 y Fo(j)2890 2902 y(n)p Fp(+1)3032 2877 y Fq(\\)t Fr(X)36 b Ft(=)28 b Fr(T)14 b(y)456 3008 y Ft(\(otherwise)24 b(if)e Fr(z)33 b Fq(2)28 b Ft(\012)1241 2961 y Fo(j)1241 3032 y(n)p Fp(+1)1383 3008 y Fq(\\)t Fr(X)j Ft(then)25 b(there)f(is)f(a)h Fr(w)30 b Fq(2)e Ft(\012)2445 2972 y Fo(i)2445 3033 y(n)2497 3008 y Fq(\\)t Fr(X)k Ft(suc)m(h)25 b(that)e Fr(T)14 b(w)30 b Ft(=)e Fr(z)t(;)456 3139 y Ft(a)k(con)m(tradiction\).)42 b(Th)m(us)34 b Fr(I)1496 3154 y Fo(T)10 b(y)1616 3139 y Ft(=)28 b(\012)1790 3091 y Fo(j)1790 3163 y(n)p Fp(+1)1927 3139 y Fr(:)555 3255 y Ft(There)40 b(are)f(only)f(a)g(\014nite)g(n)m(um)m(b)s(er)h(of)f (discon)m(tin)m(uit)m(y)h(p)s(oin)m(ts.)61 b(Th)m(us)40 b(for)e(a)456 3371 y(\014nite)44 b(n)m(um)m(b)s(er)h(of)e Fr(y)51 b Fq(2)e Fr(O)s Ft(\()p Fr(x)p Ft(\))43 b(w)m(e)j(can)f(ha)m(v) m(e)g(a)f(discon)m(tin)m(uit)m(y)h(p)s(oin)m(t)e(in)h Fr(I)3376 3386 y Fo(y)3417 3371 y Fr(:)456 3487 y Ft(In)g(particular)f (there)i(is)f(a)h Fr(n)1574 3502 y Fp(0)1658 3487 y Ft(so)g(that)f(if)f Fr(n)48 b Fq(\025)h Fr(n)2404 3502 y Fp(0)2488 3487 y Ft(then)c Fr(I)2765 3502 y Fo(T)2816 3483 y Fd(n)2859 3502 y Fo(x)2947 3487 y Ft(con)m(tains)f(no)456 3604 y(discon)m(tin)m(uit)m(y)36 b(p)s(oin)m(ts.)53 b(Th)m(us)38 b(all)c(these)j(in)m(terv)-5 b(als)35 b(ha)m(v)m(e)j(the)e(same)g (length,)h(a)456 3720 y(con)m(tradiction.)p Fw(q.e.d.)1560 3941 y Ft(5.)55 b Fs(Minimal)38 b(sets)555 4126 y Ft(Let)f Fr(D)f Ft(=)e Fq(f)p Fr(\014)1066 4141 y Fp(0)1105 4126 y Fr(;)17 b(:)g(:)g(:)f(;)h(\014)1379 4141 y Fo(r)r Fn(\000)p Fp(1)1507 4126 y Fq(g)p Fr(:)36 b Ft(F)-8 b(or)35 b(an)h(in)m(terv)-5 b(al)35 b Fr(J)46 b Ft(let)2556 4047 y Fn(\016)2537 4135 y Fr(J)2636 4126 y Ft(b)s(e)37 b(it's)e(in)m(terior)g(and)456 4279 y Fr(D)s Ft(\()p Fr(J)9 b Ft(\))32 b(=)h Fq(f)p Fr(\014)925 4294 y Fo(i)985 4279 y Fq(2)h Fr(D)h Ft(:)e Fr(T)1332 4242 y Fo(n)1379 4279 y Fr(\014)1434 4294 y Fo(i)1495 4279 y Fq(2)1612 4200 y Fn(\016)1594 4288 y Fr(J)1689 4279 y Ft(for)f(some)h Fr(n)g Fq(\025)g Ft(0)p Fq(g)p Fr(:)i Ft(W)-8 b(e)36 b(sa)m(y)h(that)e Fr(J)42 b Fq(\032)33 b Fr(I)44 b Ft(is)35 b(a)456 4395 y(minimal)30 b(in)m(terv)-5 b(al)33 b(if)h Fr(D)s Ft(\()p Fr(J)9 b Ft(\))30 b Fq(6)p Ft(=)h Fq(;)k Ft(and)f Fr(D)s Ft(\()p Fr(J)9 b Ft(\))31 b(=)g Fr(D)s Ft(\()p Fr(J)2456 4410 y Fp(1)2495 4395 y Ft(\))j(for)g(ev)m(ery)j(subin)m(terv)-5 b(al)456 4547 y Fr(J)510 4562 y Fp(1)577 4547 y Fq(\032)28 b Fr(J)41 b Ft(with)1018 4468 y Fn(\016)1000 4556 y Fr(J)1085 4547 y Fq(\\)22 b Fr(X)36 b Fq(6)p Ft(=)27 b Fq(;)p Ft(.)456 4726 y Fw(Lemma)37 b(5.1.)49 b Fk(Supp)-5 b(ose)37 b Fr(T)52 b Fk(is)38 b(ap)-5 b(erio)g(dic)38 b(and)f Fr(J)48 b Fk(is)38 b(a)g(minimal)f(interval)h(for)456 4842 y Fr(T)14 b Fk(.)44 b(Then)34 b(we)h(c)-5 b(onstruct)35 b(a)f(set)h Fr(F)49 b Fk(satisfying)34 b(the)h(fol)5 b(lowing:)548 4983 y Ft(1.)49 b Fr(J)36 b Fq(\032)29 b Fr(F)548 5099 y Ft(2.)49 b Fk(Ther)-5 b(e)34 b(is)g(an)h Fr(N)45 b Fk(such)35 b(that)g Fr(T)14 b(F)41 b Fq(\032)28 b Fr(F)36 b Fq([)23 b Ft(\()p Fr(I)30 b Fq(n)22 b Ft(\012)2462 5114 y Fo(N)2529 5099 y Ft(\))548 5216 y(3.)49 b Fr(F)f Fk(is)35 b(a)f(\014nite)h(union)f(of)h(intervals)p eop 6 5 bop 456 253 a Fl(6)524 b(J)1068 234 y(\177)1057 253 y(OR)n(G)33 b(SCHMELING)h(AND)e(SER)n(GE)h(TR)n(OUBETZK)n(O)n(Y)548 425 y Ft(4.)49 b Fr(F)35 b Fq(\\)23 b Fr(D)30 b Ft(=)e Fr(D)s Ft(\()p Fr(J)9 b Ft(\))548 541 y(5.)49 b Fr(@)5 b(F)50 b Fq(\\)36 b Fr(X)69 b Fq(\032)62 b Fr(D)1317 505 y Fn(1)1392 541 y Ft(\()p Fr(J)9 b Ft(\))61 b(=)g Fr(O)1807 505 y Fp(+)1866 541 y Ft(\()p Fr(D)s Ft(\()p Fr(J)9 b Ft(\)\))35 b Fq([)h(f)p Ft(1)p Fq(g)52 b Fk(and)h Fr(@)5 b(F)76 b Fq(\032)62 b Fr(D)3129 505 y Fn(1)3203 541 y Ft(\()p Fr(J)9 b Ft(\))36 b Fq([)673 583 y Fe(S)756 687 y Fo(k)r(>)p Fp(0)905 658 y Fq([)971 673 y Fo(\014)1011 683 y Fd(i)1037 673 y Fn(2)p Fo(D)r Fp(\()p Fo(J)6 b Fp(\))1248 658 y Fr(T)1319 622 y Fn(\000)p Fo(k)1416 658 y Fr(\014)1471 673 y Fo(i)555 834 y Fk(such)43 b(that)g Fr(X)1073 849 y Fo(F)1173 834 y Ft(:=)f Fr(X)1399 849 y Fo(J)1490 834 y Ft(:=)g Fq(\\)1701 849 y Fo(n)p Fn(\025)p Fp(0)1839 834 y Fr(T)1910 797 y Fo(n)1957 834 y Fr(F)56 b Fk(is)42 b(a)h(minimal)e(set.)68 b(Mor)-5 b(e)g(over)43 b(for)456 987 y(any)34 b Fr(J)696 1002 y Fp(1)763 987 y Fq(\032)29 b Fr(J)9 b Fk(;)1015 908 y Fn(\016)996 996 y Fr(J)1060 1002 y Fp(1)1121 987 y Fq([)23 b Fr(X)35 b Fq(6)p Ft(=)28 b Fq(;)34 b Fk(holds)h Fr(X)1845 1002 y Fo(J)1884 1011 y Fc(1)1949 987 y Ft(=)28 b Fr(X)2134 1002 y Fo(J)2211 987 y Ft(=)f Fq([)2380 1003 y Fo(\014)2420 1013 y Fd(i)2446 1003 y Fn(2)p Fo(D)r Fp(\()p Fo(J)6 b Fp(\))2657 987 y Fr(!)t Ft(\()p Fr(\014)2815 1002 y Fo(i)2843 987 y Ft(\))p Fk(.)456 1206 y Fw(Remark:)59 b Ft(In)41 b(fact)g(w)m(e)g(sho)m(w)h(a)f(stronger)g(fact)f(that)h(for) f(ev)m(ery)j Fr(x)f Fq(2)g Fr(F)54 b Ft(the)456 1323 y(orbit)31 b Fr(O)772 1286 y Fp(+)830 1323 y Ft(\()p Fr(x)p Ft(\))i(of)f Fr(x)h Ft(is)f(dense)i(in)e Fr(X)1752 1338 y Fo(F)1811 1323 y Fr(:)456 1439 y Fw(Pro)s(of)42 b(of)h(lemma:)51 b Ft(The)39 b(pro)s(of)d(uses)j(some)e(ideas)g(of)f (Keane)i([2].)57 b(Let)38 b Fr(J)45 b Ft(=)456 1555 y([)p Fr(a;)17 b(b)p Ft(\))p Fr(:)33 b Ft(Since)g Fr(J)42 b Ft(is)33 b(a)g(minimal)28 b(in)m(terv)-5 b(al)32 b(and)h Fr(D)2255 1519 y Fn(1)2358 1555 y Ft(=)28 b Fr(D)2546 1519 y Fn(1)2621 1555 y Ft(\()p Fr(J)9 b Ft(\))33 b(is)f(coun)m(table)h (w)m(e)456 1671 y(can)42 b(assume)g(that)g Fr(a)i Fq(2)h Fr(X)36 b Fq(n)28 b Fr(D)1696 1635 y Fn(1)1813 1671 y Ft(and)42 b Fr(b)i Fq(2)g Fr(X)37 b Fq(n)28 b Fr(D)2487 1635 y Fn(1)2562 1671 y Fr(;)42 b Ft(b)m(y)h(making)d Fr(J)51 b Ft(a)42 b(bit)456 1788 y(smaller)30 b(if)i(necessary)-8 b(.)555 1904 y(F)g(or)32 b(eac)m(h)h Fr(y)e Fq(2)d Fr(D)s Ft(\()p Fr(J)9 b Ft(\))22 b Fq([)h(f)p Fr(a;)17 b(b)p Fq(g)p Ft(,)32 b(set)1259 2139 y Fr(k)s Ft(\()p Fr(y)t Ft(\))27 b(=)g(inf)6 b Fq(f)p Fr(n)28 b Fq(\025)g Ft(0)g(:)f Fr(T)2132 2098 y Fo(n)2179 2139 y Fr(y)k Fq(2)d Ft(\()p Fr(a;)17 b(b)p Ft(\))p Fq(g)p Fr(:)555 2378 y Ft(The)43 b(p)s(oin)m(ts)f Fq(f)p Fr(T)1189 2342 y Fo(k)r Fp(\()p Fo(y)r Fp(\))1324 2378 y Fr(y)47 b Ft(:)d Fr(y)j Fq(2)e Fr(D)31 b Fq([)e(f)p Fr(a;)17 b(b)p Fq(g)p Fr(;)g(k)s Ft(\()p Fr(y)t Ft(\))43 b Fr(<)h Fq(1)p Ft(\))p Fq(g)e Ft(partition)e Fr(J)51 b Ft(in)m(to)456 2494 y(in)m(terv)-5 b(als)45 b Fr(J)914 2509 y Fp(1)954 2494 y Fr(;)17 b(:)g(:)g(:)e(;)i(J) 1226 2509 y Fo(l)1298 2494 y Ft(where)48 b Fr(l)54 b Fq(\024)d Fr(r)35 b Ft(+)c(2)p Fr(:)46 b Ft(Let)h Fr(N)57 b Ft(b)s(e)47 b(the)g(smallest)d(in)m(teger)456 2610 y(whic)m(h)35 b(is)f(larger)g(than)g(max\()p Fq(j)p Fr(J)1647 2625 y Fo(i)1675 2610 y Fq(j)1703 2574 y Fn(\000)p Fp(1)1828 2610 y Ft(:)d(1)g Fq(\024)h Fr(i)g Fq(\024)f Fr(l)r Ft(\))p Fr(:)k Ft(In)g(the)g(next)h(argumen)m(t)e(w)m(e)456 2726 y(will)c(only)i(need)i(to)e(consider)h Fr(T)1652 2690 y Fn(\000)p Fo(n)1786 2726 y Ft(for)f(0)27 b Fq(\024)h Fr(n)g Fq(\024)g Fr(N)5 b(:)555 2843 y Ft(F)-8 b(or)40 b(0)h Fq(\024)g Fr(n)h Fq(\024)f Fr(N)51 b Ft(the)41 b(cardinalit)m(y)e(of)h(the)h(set)g Fr(T)2491 2806 y Fn(\000)p Fo(n)2592 2843 y Fr(x)g Ft(is)f(b)s(ounded)i(ab)s(o)m(v)m(e) 456 2959 y(b)m(y)37 b(a)g(constan)m(t)h(\(dep)s(ending)f(on)f Fr(N)10 b Ft(\).)57 b(If)37 b(w)m(e)h(consider)f(the)g(preimage)e(tree) j(of)456 3075 y(length)32 b Fr(N)44 b Ft(w)m(e)34 b(can)f(lab)s(el)e (eac)m(h)j(of)f(the)g(\014nite)g(branc)m(hes)i(b)m(y)f(a)e(subscript)i Fr(i)p Ft(,)g(i.e.)456 3191 y Fr(T)527 3150 y Fn(\000)p Fo(n)513 3217 y(i)628 3191 y Fr(x)f Ft(is)f(a)h(c)m(hoice)f(of)h (preimage)e(branc)m(hes.)555 3308 y(Set)42 b Fr(n)p Ft(\()p Fr(i;)17 b(j)6 b Ft(\))42 b(to)f(b)s(e)g(the)h(smallest)e Fr(n)i Ft(with)f(0)h Fr(<)h(n)g(<)f(N)52 b Ft(suc)m(h)43 b(that)f(either)456 3460 y Fr(T)527 3419 y Fn(\000)p Fo(n)513 3485 y(i)647 3381 y Fn(\016)628 3469 y Fr(J)691 3475 y Fo(j)755 3460 y Fq(\\)27 b Fr(J)49 b Fq(6)p Ft(=)39 b Fq(;)g Ft(or)g Fr(T)1352 3419 y Fn(\000)p Fo(n)1338 3485 y(i)1472 3381 y Fn(\016)1454 3469 y Fr(J)1517 3475 y Fo(j)1593 3460 y Ft(=)g Fq(;)p Ft(.)64 b(A)40 b(priori)d(it)i(is)g (not)g(clear)g(that)g Fr(n)p Ft(\()p Fr(i;)17 b(j)6 b Ft(\))40 b(is)456 3576 y(de\014ned)f(for)e(all)f Fr(i;)17 b(j:)38 b Ft(T)-8 b(o)38 b(see)h(this)e(note)h(that)g(if)f Fr(m)f(<)h(n)p Ft(\()p Fr(i;)17 b(j)6 b Ft(\))38 b(\(i.e.)58 b(the)38 b(orbit)456 3729 y(of)33 b Fr(J)622 3744 y Fo(j)693 3729 y Ft(has)h(not)g(y)m(et)i(returned)f(to)f Fr(J)9 b Ft(\))34 b(then)h Fr(T)2156 3687 y Fn(\000)p Fo(m)2142 3754 y(i)2295 3650 y Fn(\016)2277 3738 y Fr(J)2340 3744 y Fo(j)2400 3729 y Fq(\\)23 b Fr(D)33 b Ft(=)e Fq(;)j Ft(b)m(y)h(de\014nition)e(of)456 3884 y(the)38 b(partition.)56 b(This)37 b(implies)e(that)j(all)d(of)i(the)h(preimages)f Fr(T)2829 3843 y Fn(\000)p Fo(m)2815 3909 y(i)2969 3805 y Fn(\016)2950 3893 y Fr(J)3013 3899 y Fo(j)3087 3884 y Ft(for)g(\014xed)456 4000 y Fr(j)6 b Ft(,)40 b(all)c Fr(i)j Ft(and)f(all)f Fr(m)h(<)g(n)p Ft(\()p Fr(i;)17 b(j)6 b Ft(\))38 b(are)h(disjoin)m(t.)60 b(W)-8 b(e)39 b(also)f(ha)m(v)m(e)h(for)f(all)f Fr(i;)17 b(j)44 b Ft(and)456 4152 y(0)29 b Fr(<)g(m)g Fq(\024)h Fr(n)p Ft(\()p Fr(i;)17 b(j)6 b Ft(\))33 b(either)g Fr(T)1498 4111 y Fn(\000)p Fo(m)1484 4178 y(i)1638 4074 y Fn(\016)1620 4161 y Fr(J)1683 4167 y Fo(j)1748 4152 y Ft(=)c Fq(;)34 b Ft(or)f Fr(T)2128 4116 y Fo(m)2194 4152 y Ft(\()p Fr(T)2303 4111 y Fn(\000)p Fo(m)2289 4178 y(i)2443 4074 y Fn(\016)2424 4161 y Fr(J)2487 4167 y Fo(j)2524 4152 y Ft(\))c(=)g Fr(J)2750 4167 y Fo(j)2786 4152 y Ft(.)47 b(Th)m(us)35 b(w)m(e)f(ha)m(v)m(e)456 4269 y(that)i(either)h Fr(J)1006 4284 y Fo(j)1078 4269 y Fq(\032)f Ft(\012)1261 4284 y Fo(N)1366 4269 y Ft(or)h Fr(J)1544 4284 y Fo(j)1605 4269 y Fq(\\)26 b Ft(\012)1767 4284 y Fo(N)1870 4269 y Ft(=)35 b Fq(;)p Fr(:)i Ft(If)g Fr(J)2251 4284 y Fo(j)2323 4269 y Fq(\032)f Ft(\012)2506 4284 y Fo(N)2611 4269 y Ft(w)m(e)i(ha)m(v)m(e)g(that)f(for)g(at)456 4385 y(least)g(one)i(of)f(the)h(branc)m(hes)h Fr(i)p Ft(\()p Fr(j)6 b Ft(\))38 b(for)g(all)e Fr(m)i Ft(suc)m(h)i(that)e(0)f Fr(<)h(m)f(<)h(n)p Ft(\()p Fr(i)p Ft(\()p Fr(j)6 b Ft(\))p Fr(;)17 b(j)6 b Ft(\))456 4501 y(the)34 b Fr(T)696 4460 y Fn(\000)p Fo(m)682 4533 y(i)p Fp(\()p Fo(j)t Fp(\))817 4501 y Fr(J)871 4516 y Fo(j)942 4501 y Ft(ha)m(v)m(e)h(the)f(same)g (length,)g Fq(j)p Fr(J)1992 4516 y Fo(j)2028 4501 y Fq(j)p Fr(:)g Ft(Since)g(they)h(are)f(disjoin)m(t)f(w)m(e)i(ha)m(v)m(e)456 4632 y Fr(n)p Ft(\()p Fr(i;)17 b(j)6 b Ft(\))p Fq(j)p Fr(J)795 4647 y Fo(j)831 4632 y Fq(j)27 b Fr(<)h Ft(1)k(and)h(th)m(us)g Fr(n)p Ft(\()p Fr(i;)17 b(j)6 b Ft(\))28 b Fr(<)f(N)44 b Ft(for)32 b(all)e Fr(i;)17 b(j:)555 4748 y Ft(Since)48 b Fr(a)g Ft(and)g Fr(b)h Ft(are)f(p)s(oin)m(ts)f(from)g Fr(X)56 b Ft(they)49 b(ha)m(v)m(e)g(an)f(in\014nite)f(preimage)456 4865 y(path)37 b(con)m(tained)h(in)f Fr(X)8 b Ft(.)59 b(In)38 b(the)g(ab)s(o)m(v)m(e)h(construction)f(w)m(e)h(will)c (disregard)i(all)456 4981 y(preimage)e(in)m(terv)-5 b(als)35 b Fr(T)1342 4945 y Fn(\000)p Fo(k)1439 4981 y Fr(J)1493 4996 y Fo(j)1566 4981 y Ft(where)i Fr(J)1905 4996 y Fo(j)1978 4981 y Ft(has)f Fr(a)h Ft(or)e Fr(b)i Ft(as)f(an)h(endp)s(oin)m(t)f (and)g(the)456 5099 y(endp)s(oin)m(t)k(of)g Fr(T)1060 5063 y Fn(\000)p Fo(k)1158 5099 y Fr(J)1212 5114 y Fo(j)1289 5099 y Ft(corresp)s(onding)g(to)h(the)g(preimage)e(of)h Fr(a)h Ft(resp)s(ectiv)m(ely)h Fr(b)456 5216 y Ft(are)32 b(not)h(in)e(some)i(preimage)e(path)i(con)m(tained)f(in)g Fr(X)8 b Ft(.)p eop 7 6 bop 1224 236 a Fl(INTER)-9 b(V)g(AL)33 b(TRANSLA)-6 b(TION)30 b(MAPPINGS)730 b(7)555 461 y Ft(F)-8 b(urthermore,)25 b Fr(T)1211 410 y Fn(\000)p Fo(n)p Fp(\()p Fo(i;j)t Fp(\))1197 487 y Fo(i)1462 382 y Fn(\016)1443 470 y Fr(J)1507 476 y Fo(j)1566 461 y Ft(can)f(not)g(con)m(tain)f Fr(a)g Ft(or)g Fr(b)p Ft(,)j(for)d(if)f(it)h(con)m(tains)g(one)h(of)456 617 y(them,)k(sa)m(y)g Fr(a)p Ft(,)g(then)g Fr(T)1285 581 y Fo(n)1331 617 y Fr(a)g Fq(2)1523 538 y Fn(\016)1504 626 y Fr(J)1567 632 y Fo(j)1604 617 y Ft(,)g(a)f(con)m(tradiction.)40 b(Th)m(us,)30 b(if)c Fr(T)2789 581 y Fn(\000)p Fo(n)p Fp(\()p Fo(i;j)t Fp(\))3022 617 y Fr(J)3076 632 y Fo(j)3123 617 y Fq(\\)11 b Fr(J)37 b Fq(6)p Ft(=)28 b Fq(;)456 754 y Ft(then)i Fr(T)746 703 y Fn(\000)p Fo(n)p Fp(\()p Fo(i;j)t Fp(\))732 779 y Fo(i)978 754 y Fr(J)1032 769 y Fo(j)1096 754 y Fq(\032)f Fr(J)n(:)g Ft(In)h(the)g(case)h Fr(J)1852 769 y Fo(j)1904 754 y Fq(\\)16 b Ft(\012)2056 769 y Fo(N)2152 754 y Ft(=)27 b Fq(;)j Ft(w)m(e)g(can)g(not)f(guaran)m (tee)i(that)456 870 y(a)24 b(single)g(branc)m(h)h(comes)g(bac)m(k)h(to) e Fr(J)1772 885 y Fo(j)1808 870 y Ft(.)41 b(None)25 b(the)g(less,)i(if) c(some)i(branc)m(h)g(returns,)456 986 y(it)g(m)m(ust)i(return)g (completely)-8 b(,)26 b(and)h(all)d(branc)m(hes)29 b(whic)m(h)e(do)f (not)g(return)h(die)f(out)456 1102 y(in)31 b(less)i(than)g Fr(N)43 b Ft(steps.)555 1218 y(Let)1230 1344 y Fr(F)e Ft(=)28 b Fr(F)14 b Ft(\()p Fr(J)9 b Ft(\))27 b(=)g Fq([)1850 1303 y Fo(l)1850 1368 y(j)t Fp(=1)2000 1344 y Fq([)2066 1359 y Fo(i)2116 1344 y Fq([)2182 1293 y Fo(n)p Fp(\()p Fo(i;j)t Fp(\))2182 1368 y Fo(m)p Fp(=0)2361 1344 y Fr(T)2432 1302 y Fn(\000)p Fo(m)2418 1369 y(i)2553 1344 y Fr(J)2607 1359 y Fo(j)2643 1344 y Fr(:)456 1493 y Ft(It)32 b(follo)m(ws)g(that)g Fr(F)46 b Ft(is)33 b(a)f(\014nite)g(union)g(of)h(in)m(terv)-5 b(als.)43 b(Also)32 b Fr(D)s Ft(\()p Fr(F)14 b Ft(\))27 b(=)g Fr(D)s Ft(\()p Fr(J)9 b Ft(\).)44 b(By)456 1610 y(the)33 b(preceding)g(construction,)g(if)f(w)m(e)i(rede\014ne)g Fr(N)44 b Ft(to)32 b(b)s(e)h(max)16 b Fr(n)p Ft(\()p Fr(i;)h(j)6 b Ft(\))33 b(w)m(e)h(ha)m(v)m(e)456 1726 y Fr(F)h Fq(n)22 b Fr(T)697 1690 y Fn(\000)p Fp(1)791 1726 y Fr(F)42 b Fq(\032)28 b Fr(I)i Fq(n)22 b Ft(\012)1216 1741 y Fo(N)1284 1726 y Ft(.)43 b(Hence,)34 b Fr(T)14 b(F)41 b Fq(\032)28 b Fr(F)36 b Fq([)23 b Ft(\()p Fr(I)30 b Fq(n)22 b Ft(\012)2392 1741 y Fo(N)7 b Fp(+1)2549 1726 y Ft(\).)555 1842 y(Next)34 b(w)m(e)f(need)h(the)f(follo)m(wing)d (lemma.)456 2022 y Fw(Lemma)37 b(5.2.)49 b Fk(Supp)-5 b(ose)36 b Fr(J)1503 2037 y Fp(1)1580 2022 y Fk(and)g Fr(J)1825 2037 y Fp(2)1902 2022 y Fk(ar)-5 b(e)37 b(minimal)e (intervals.)51 b(Then)36 b(exactly)456 2138 y(one)e(of)g(the)h(fol)5 b(lowing)34 b(two)h(options)f(holds)548 2279 y Ft(1.)49 b Fr(D)s Ft(\()p Fr(J)849 2294 y Fp(1)888 2279 y Ft(\))27 b(=)h Fr(D)s Ft(\()p Fr(J)1233 2294 y Fp(2)1272 2279 y Ft(\))35 b Fk(and)f Fr(X)1615 2294 y Fo(J)1654 2303 y Fc(1)1720 2279 y Ft(=)27 b Fr(X)1904 2294 y Fo(J)1943 2303 y Fc(2)2009 2279 y Ft(=)h Fq([)2179 2294 y Fo(\014)2219 2304 y Fd(i)2245 2294 y Fn(2)p Fo(D)r Fp(\()p Fo(J)2418 2303 y Fc(1)2453 2294 y Fp(\))2484 2279 y Fr(!)t Ft(\()p Fr(\014)2642 2294 y Fo(i)2670 2279 y Ft(\))p Fr(:)548 2395 y Ft(2.)49 b Fr(D)s Ft(\()p Fr(J)849 2410 y Fp(1)888 2395 y Ft(\))22 b Fq(\\)g Fr(D)s Ft(\()p Fr(J)1212 2410 y Fp(2)1251 2395 y Ft(\))28 b(=)f Fq(;)35 b Fk(and)f Fr(X)1775 2410 y Fo(J)1814 2419 y Fc(1)1875 2395 y Fq(\\)22 b Fr(X)2044 2410 y Fo(J)2083 2419 y Fc(2)2149 2395 y Ft(=)28 b Fq(;)555 2575 y Ft(If)h Fr(J)712 2538 y Fn(0)764 2575 y Fq(2)f Fr(T)929 2538 y Fo(n)975 2575 y Fr(F)43 b Ft(is)29 b(an)g(in)m(terv)-5 b(al)28 b(whose)i(in)m(terior)e(has)i (non{empt)m(y)f(in)m(tersection)456 2691 y(with)22 b Fr(X)31 b Ft(then)23 b Fr(J)1055 2655 y Fn(0)1101 2691 y Ft(is)f(minimal.)37 b(Th)m(us)24 b(the)f(lemma)e(sho)m(ws)j(that)f (for)f(eac)m(h)i(in)m(terv)-5 b(al)456 2807 y(in)m(tersecting)30 b Fr(X)38 b Ft(w)m(e)31 b(ha)m(v)m(e)g Fr(D)s Ft(\()p Fr(J)1644 2771 y Fn(0)1667 2807 y Ft(\))d(=)f Fr(D)s Ft(\()p Fr(J)9 b Ft(\))30 b(so)h(that)e Fr(X)2496 2822 y Fo(J)2573 2807 y Fq(\032)f([)2744 2823 y Fo(\014)2784 2833 y Fd(i)2811 2823 y Fn(2)p Fo(D)r Fp(\()p Fo(J)6 b Fp(\))3021 2807 y Fr(!)t Ft(\()p Fr(\014)3179 2822 y Fo(i)3207 2807 y Ft(\).)42 b(On)456 2923 y(the)22 b(other)h(hand,)i (since)e Fr(F)36 b Ft(con)m(tains)22 b(at)h(least)f(one)g(orbit)g(p)s (oin)m(t)f(of)h(all)f Fr(\014)3072 2938 y Fo(i)3128 2923 y Fq(2)28 b Fr(D)s Ft(\()p Fr(J)9 b Ft(\))456 3040 y(and)32 b(clearly)g Fr(!)t Ft(\()p Fr(\014)1115 3055 y Fo(i)1142 3040 y Ft(\))c Fq(\032)g Ft(\()p Fr(I)i Fq(n)22 b Ft(\012)1566 3055 y Fo(n)1613 3040 y Ft(\))33 b(for)f(all)e Fr(n)j Ft(w)m(e)h(conclude)1515 3205 y Fr(X)1596 3220 y Fo(j)1660 3205 y Ft(=)27 b Fq([)1829 3221 y Fo(\014)1869 3231 y Fd(i)1896 3221 y Fn(2)p Fo(D)r Fp(\()p Fo(J)2069 3230 y Fc(1)2103 3221 y Fp(\))2135 3205 y Fr(!)t Ft(\()p Fr(\014)2293 3220 y Fo(i)2320 3205 y Ft(\))p Fr(:)555 3371 y Ft(If)40 b Fr(x)g Ft(is)f(a)g(b)s(oundary)h(p)s(oin)m(t)f(of)g Fr(F)53 b Ft(not)39 b(b)s(elonging)f(to)h Fr(D)2729 3335 y Fn(1)2804 3371 y Ft(,)i(then)f(for)f(eac)m(h)456 3487 y Fr(m)45 b Fq(\025)h Ft(0)d Fr(T)872 3451 y Fn(\000)p Fo(m)1036 3487 y Ft(is)g(con)m(tin)m(uous)g(at)g Fr(x)p Ft(,)j(pro)m(vided)e(this)e(map)g(is)h(de\014ned.)76 b(This)456 3604 y(implies)36 b(that)j Fr(T)1082 3568 y Fn(\000)p Fp(1)1215 3604 y Ft(m)m(ust)h(\014x)g(the)f(part)g(of)g (the)h(b)s(oundary)f(of)g Fr(F)53 b Ft(whic)m(h)39 b(is)g(in)456 3720 y Fr(X)26 b Fq(n)18 b Fr(D)715 3684 y Fn(1)790 3720 y Fr(;)31 b Ft(ho)m(w)m(ev)m(er)i(since)e(this)g(is)f(a)h(\014nite)f (set)i(this)e(con)m(tradicts)i(the)f(fact)g(that)456 3836 y Fr(O)534 3800 y Fp(+)592 3836 y Ft(\()p Fr(x)p Ft(\))43 b(is)f(in\014nite)g(for)h(all)d Fr(x)p Ft(.)75 b(Th)m(us)44 b(the)g(b)s(oundary)f(of)f Fr(F)57 b Ft(cannot)43 b(con)m(tain)456 3952 y Fr(a)36 b Ft(or)f Fr(b)h Ft(or)g(preimages)f (of)g(them)h(\(Note)g(that)f(w)m(e)i(considered)g(only)e(particular)456 4069 y(preimage)27 b(in)m(terv)-5 b(als)28 b(for)g Fr(a)h Ft(and)g Fr(b)p Ft(.\).)42 b(The)30 b(other)f(endp)s(oin)m(ts)g(are)g (all)d(preimages)456 4185 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Ft(+)667 1408 y Fe(P)772 1434 y Fo(k)772 1512 y(n)p Fp(=1)926 1483 y Fr(\015)977 1498 y Fo(j)1006 1506 y Fd(n)1052 1483 y Ft(.)62 b(Finally)-8 b(,)37 b(w)m(e)j(de\014ne)g (blossoms.)61 b(These)40 b(are)f(ob)5 b(jects)40 b(whic)m(h)456 1599 y(are)28 b(sp)s(eci\014c)i(to)e Fr(T)42 b Ft(but)29 b(in)f(con)m(trary)i(to)e(fertile)f(paths)j(c)m(hange)f(with)g(time)e Fr(n)p Ft(.)42 b(A)456 1715 y(blossom)29 b(at)i(time)e Fr(n)i Ft(is)f(a)g(mark)m(ed)h(v)m(ertex.)45 b(W)-8 b(e)31 b(mark)f(a)g(v)m(ertex)j(\()p Fr(i;)17 b(j)3047 1730 y Fp(1)3087 1715 y Fr(;)g Fq(\001)g(\001)g(\001)31 b Fr(;)17 b(j)3364 1730 y Fo(k)3406 1715 y Ft(\))456 1843 y(at)38 b(time)g Fr(n)h Ft(i\013)e(it)h(lies)g(on)h(a)f(fertile)g(path) h(and)f Fr(T)2323 1807 y Fo(k)2366 1843 y Fr(\014)2421 1858 y Fo(i)2487 1843 y Ft(=)h Fr(\014)2657 1858 y Fo(i)2711 1843 y Ft(+)2814 1768 y Fe(P)2919 1795 y Fo(k)2919 1872 y(n)p Fp(=1)3073 1843 y Fr(\015)3124 1858 y Fo(j)3153 1866 y Fd(n)3237 1843 y Ft(is)f(an)456 1959 y(endp)s(oin)m(t)33 b(of)g(an)h(in)m(terv)-5 b(al)32 b(in)h(\012)1650 1974 y Fo(n)1697 1959 y Ft(.)47 b(Here)34 b(w)m(e)g(consider)g(the)g (singularities)d Fr(\014)3295 1974 y Fo(i)3357 1959 y Ft(as)456 2075 y(endp)s(oin)m(ts)g(to)s(o.)43 b(W)-8 b(e)31 b(w)m(an)m(t)i(to)e(remark)g(that)g(for)g(blossoms)g(the)h (distance)f Fr(k)k Ft(to)456 2192 y(the)c(ro)s(ot)f(of)g(the)h(tree)h (is)e(alw)m(a)m(ys)h(less)g(than)g(or)g(equal)f(to)h(the)g(time)f(of)g (marking)456 2308 y Fr(n)p Ft(.)42 b(So)29 b(the)g(ev)m(olution)f(of)h (the)g(cutting)f(park)h(c)m(haracterizes)h(the)g(com)m(binatorial)456 2424 y(structure)37 b(of)e(an)h(ITM.)h(It)f(is)g(clear)f(that)h(the)h (ev)m(olution)e(of)g(a)h(cutting)f(park's)456 2540 y(blossoms)h(stops)i (at)e(a)h(\014nite)f(time)g Fr(n)h Ft(if)e(and)i(only)f(if)g Fr(T)51 b Ft(is)36 b(of)g(\014nite)h(t)m(yp)s(e.)57 b(In)456 2657 y(fact)37 b(it)g(is)h(of)f(\014nite)h(t)m(yp)s(e)h(if)e(and)h (only)f(if)g(the)i(blossoms)e(at)h(t)m(w)m(o)h(consecutiv)m(e)456 2773 y(time)g(steps)j(are)e(the)h(same.)67 b(A)41 b(v)m(ery)h(imp)s (ortan)m(t)c(observ)-5 b(ation)40 b(is)g(that)h(b)s(oth)456 2889 y(marking)33 b(the)j(arro)m(ws)g(or)e(marking)g(the)i(v)m(ertices) g(is)f(de\014ned)h(at)f(a)g(\014nite)g(time)456 3005 y(\(lev)m(el\))d(b)m(y)h(not)g(more)e(than)i(\()p Fr(n)22 b Ft(+)g(1\))1837 2969 y Fp(2)1877 3005 y Fr(=)p Ft(2)32 b(inequalities)e(of)i(the)h(form)1410 3260 y Fr(\014)1465 3275 y Fo(i)1516 3260 y Ft(+)1650 3135 y Fo(k)1687 3144 y Fc(1)1614 3166 y Fe(X)1630 3378 y Fo(l)q Fp(=1)1774 3260 y Fr(\015)1825 3275 y Fo(i)1849 3287 y Fd(l)1905 3260 y Fq(\024)28 b Fr(\014)2065 3275 y Fo(j)2124 3260 y Ft(+)2258 3135 y Fo(k)2295 3144 y Fc(2)2222 3166 y Fe(X)2238 3378 y Fo(l)q Fp(=1)2382 3260 y Fr(\015)2433 3275 y Fo(j)2462 3287 y Fd(l)3320 3260 y Ft(\(1\))456 3506 y(with)34 b Fr(k)731 3521 y Fp(1)770 3506 y Fr(;)17 b(k)865 3521 y Fp(2)936 3506 y Fq(\024)32 b Fr(n)p Ft(,)j(1)d Fq(\024)g Fr(i;)17 b(j)37 b Fq(\024)32 b Fr(r)38 b Ft(and)d(1)c Fq(\024)h Fr(i)2114 3521 y Fo(l)2140 3506 y Fr(;)17 b(j)2224 3521 y Fo(l)2282 3506 y Fq(\024)32 b Fr(r)s Ft(.)50 b(These)36 b(inequalities)d(are)456 3622 y(called)e(the)i(de\014ning)g (inequalities)d(at)i(time)g Fr(n)p Ft(.)456 3792 y Fw(Lemma)37 b(8.1.)49 b Fk(L)-5 b(et)43 b Fr(T)56 b Fk(have)41 b(maximal)g(r)-5 b(ank.)67 b(Then)42 b(for)g(e)-5 b(ach)42 b Fr(n)g Fk(ther)-5 b(e)42 b(is)h(a)456 3908 y(neighb)-5 b(orho)g(o)g(d)31 b(of)h Fr(T)47 b Fk(such)32 b(that)i(al)5 b(l)32 b(ITM's)g(form)g(this) h(neighb)-5 b(orho)g(o)g(d)31 b(have)h(the)456 4024 y(same)i(cutting)h (p)-5 b(ark)34 b(up)h(to)g(time)g Fr(n)g Fk(as)g Fr(T)14 b Fk(.)456 4275 y Fw(Pro)s(of)42 b(of)g(the)h(lemma)d(8.1:)53 b Ft(If)37 b(all)d(the)k(inequalities)d(in)h(1)g(are)h(strict)g(then) 456 4391 y(the)c(assertion)f(follo)m(ws)g(immediately)d(since)k(there)h (are)e(only)g(\014nitely)g(man)m(y)h(of)456 4507 y(these)27 b(conditions.)41 b(If)26 b(some)h(of)f(these)h(conditions)f(in)g(1)g (are)g(actually)g(equalities)456 4624 y(w)m(e)33 b(conclude)g(as)g (follo)m(ws.)42 b(Let)1398 4854 y Fr(\014)1453 4869 y Fo(i)1503 4854 y Ft(+)1637 4729 y Fo(k)1674 4738 y Fc(1)1601 4759 y Fe(X)1617 4971 y Fo(l)q Fp(=1)1761 4854 y Fr(\015)1812 4869 y Fo(i)1836 4881 y Fd(l)1892 4854 y Ft(=)28 b Fr(\014)2051 4869 y Fo(j)2109 4854 y Ft(+)2244 4729 y Fo(k)2281 4738 y Fc(2)2207 4759 y Fe(X)2223 4971 y Fo(l)q Fp(=1)2368 4854 y Fr(\015)2419 4869 y Fo(j)2448 4881 y Fd(l)2475 4854 y Fr(:)818 b Ft(\(2\))456 5099 y(Since)38 b Fr(T)51 b Ft(has)39 b(maximal)c(rank)j(this)g(is)f(only)h(p)s(ossible)f(if)g Fr(k)2675 5114 y Fp(1)2751 5099 y Ft(=)g Fr(k)2915 5114 y Fp(2)2954 5099 y Ft(,)i Fr(i)f Ft(=)e Fr(j)44 b Ft(and)456 5216 y(there)23 b(is)g(a)g(p)s(erm)m(utation)f Fr(\033)27 b Ft(of)22 b(1)p Fr(;)17 b Fq(\001)g(\001)g(\001)31 b Fr(;)17 b(k)1925 5231 y Fp(1)1987 5216 y Ft(with)23 b Fr(i)2233 5231 y Fo(\033)r Fp(\()p Fo(l)q Fp(\))2384 5216 y Ft(=)28 b Fr(j)2528 5231 y Fo(l)2554 5216 y Ft(.)40 b(No)m(w)24 b(if)e(w)m(e)i(c)m(hange)g Fr(\014)3416 5231 y Fo(i)p eop 12 11 bop 456 253 a Fl(12)486 b(J)1068 234 y(\177)1057 253 y(OR)n(G)33 b(SCHMELING)h(AND)e(SER)n(GE)h(TR)n(OUBETZK)n(O)n(Y)456 425 y Ft(to)22 b Fr(\014)626 389 y Fn(0)620 450 y Fo(i)671 425 y Ft(and)h Fr(\015)902 440 y Fo(j)961 425 y Ft(to)f Fr(\015)1126 389 y Fn(0)1121 450 y Fo(j)1180 425 y Ft(b)s(oth)g(sums)h (in)f(2)g(c)m(hange)i(for)e(the)h(same)f(amoun)m(t)g(and)h(hence,)456 541 y(equalit)m(y)37 b(is)g(preserv)m(ed.)62 b(Therefore)38 b(the)g(system)h(of)e(de\014ning)h(inequalities)e(is)456 658 y(preserv)m(ed)f(in)d(a)g(neigh)m(b)s(orho)s(o)s(d)f(of)h Fr(T)14 b Ft(.)p Fw(q.e.d.)456 855 y(Pro)s(of)35 b(of)i(theorem)e(2.9:) 43 b Ft(Let)32 b Fr(T)45 b Ft(b)s(e)31 b(of)g(maximal)d(rank)k(and)f (of)g(\014nite)g(t)m(yp)s(e.)456 971 y(This)c(means)h(that)g(the)g (cutting)f(park)h(do)s(es)g(not)f(c)m(hange)i(an)m(y)f(more)f(after)g (some)456 1088 y(time)34 b Fr(n)p Ft(.)54 b(By)37 b(lemma)d(8.1)h (there)i(is)f(a)f(neigh)m(b)s(orho)s(o)s(d)g(of)h Fr(T)50 b Ft(suc)m(h)37 b(that)f(all)e(the)456 1204 y(ITM's)46 b(in)f(the)h(neigh)m(b)s(orho)s(o)s(d)e(ha)m(v)m(e)i(the)g(same)f (cutting)g(park)h(as)f Fr(T)59 b Ft(up)46 b(to)456 1320 y(time)27 b Fr(n)15 b Ft(+)g(1)p Fr(:)28 b Ft(Th)m(us)i(since)g(the)f (blossoms)f(of)h(the)g(cutting)f(parks)i(do)f(not)f(c)m(hange)456 1436 y(from)35 b(time)h Fr(n)h Ft(to)g Fr(n)25 b Ft(+)g(1)37 b(all)e(the)i(ITM's)i(in)d(this)g(neigh)m(b)s(orho)s(o)s(d)g(are)h(of)g (\014nite)456 1553 y(t)m(yp)s(e.)p Fw(q.e.d.)1660 1750 y Fs(References)497 1908 y Fv([1])k(M.)27 b(Boshernitzan)f(and)h(I.)g (Kornfeld)f Fm(Interval)j(tr)l(anslation)h(mappings)e Fv(Erg.)e(Th.)h(Dyn.)626 2008 y(Sys.)h(\(1995\))e(821{831.)497 2107 y([2])41 b(M.)28 b(Keane)f Fm(Interval)j(exchange)g(tr)l (ansformations)f Fv(Math.)e(Z.)h(141)e(\(1975\))g(25{31.)555 2364 y Fu(Freie)50 b(Universit)1237 2357 y(\177)1234 2364 y(at)f(Berlin,)54 b(FB)49 b(Ma)-6 b(thema)g(tik)50 b(und)f(Inf)n(orma)-6 b(tik,)53 b(Arni-)456 2463 y(mallee)31 b(2-6,)g(D{14195)f(Berlin)555 2563 y Fm(E-mail)h(addr)l(ess)7 b Fv(:)38 b Fa(schmeling@math.f)o(u-)o(ber)o(li)o(n.d)o(e)555 2732 y Fu(Dep)-6 b(ar)g(tment)27 b(of)e(Ma)-6 b(thema)g(tics,)27 b(University)e(of)g(Alabama)e(a)-6 b(t)26 b(Birmingham,)456 2832 y(Birmingham,)31 b(AL)f(35294)555 2932 y Fm(E-mail)h(addr)l(ess)7 b Fv(:)38 b Fa(troubetz@math.ua)o(b.)o(edu)p eop end userdict /end-hook known{end-hook}if