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b(h)m(yp)s(erb)s(olic)k(p)s(erio)s(dic)f(p)s(oin)m(ts)h(are)g(dense)i (in)e Fr(M)38 b Ft(\(with)27 b(dimension)f(higher)-236 2218 y(than)j(2\))f(and)h(one)g(of)f(them)g(has)h(co)s(dimension)f (one,)h(then,)i(the)e(homeomorphism)d(is)i(conjugated)h(to)g(a)f (linear)-236 2358 y(Anoso)m(v)34 b(di\013eomorphism)e(of)h Fo(T)975 2315 y Fm(n)1022 2358 y Ft(.)47 b(The)34 b(reason)g(wh)m(y)h (this)f(do)s(es)g(not)f(w)m(ork)i(in)e(dimension)f(2)h(is)g(that)h(w)m (e)g(can)-236 2497 y(disconnect)i(an)g(arc)g(b)m(y)g(remo)m(ving)f (from)f(it)h(one)h(p)s(oin)m(t)f(and)h(not)f(a)g(disc)h(of)f(dimension) g Fr(>)d Ft(1.)53 b(The)37 b(pro)s(of)d(in)-236 2637 y(this)c(case)h(is)e(based)i(on)f(pro)m(ving)g(\014rst)h(that)f (singularities)d(are)k(\014nite,)f(and)g(then)h(discarding)e(their)h (existence.)-236 2802 y SDict begin H.S end -236 2802 a -236 2802 a SDict begin 16.79996 H.A end -236 2802 a -236 2802 a SDict begin [/View [/XYZ H.V]/Dest (subsection.1.1) cvn /DEST pdfmark end -236 2802 a 120 x Ft(1.1.)48 b Fy(De\014nitions)39 b(and)i(presen)m(tation)f(of)g(results.)48 b Ft(In)35 b(this)g(section)g(w)m(e)g(de\014ne)h(the)g(concepts)g(that)f(w)m(e) -236 3062 y(use)e(in)f(the)h(course)h(of)e(this)g(pap)s(er)h(and)f(giv) m(e)h(precise)g(statemen)m(ts)h(of)e(the)h(results)g(in)f(it.)-236 3301 y SDict begin H.S end -236 3301 a -236 3301 a SDict begin 16.79996 H.A end -236 3301 a -236 3301 a SDict begin [/View [/XYZ H.V]/Dest (defi.1.1) cvn /DEST pdfmark end -236 3301 a Fy(De\014nition)40 b(1.1.)j Fl(We)38 b(say)g(an)g(home)-5 b(omorphism)35 b Fr(f)22 b Ft(:)34 b Fr(M)44 b Fq(!)33 b Fr(M)49 b Fl(is)d Ft(expansiv)m(e)39 b Fl(if)f Fr(\013)c(>)f Ft(0)38 b Fl(exists)f(satisfying)-236 3441 y(that)e(if)g Fr(x;)17 b(y)31 b Fq(2)d Fr(M)45 b Fl(ar)-5 b(e)35 b(di\013er)-5 b(ent)34 b(p)-5 b(oints,)35 b(then,)f(ther)-5 b(e)35 b(exists)g Fr(n)27 b Fq(2)i Fo(Z)j Fl(such)i(that)i(dist)p Ft(\()p Fr(f)3046 3405 y Fm(n)3092 3441 y Ft(\()p Fr(x)p Ft(\))p Fr(;)17 b(f)3326 3405 y Fm(n)3373 3441 y Ft(\()p Fr(y)t Ft(\)\))27 b Fr(>)g(\013)q Fl(.)-236 3680 y SDict begin H.S end -236 3680 a -236 3680 a SDict begin 16.79996 H.A end -236 3680 a -236 3680 a SDict begin [/View [/XYZ H.V]/Dest (defi.1.2) cvn /DEST pdfmark end -236 3680 a Fy(De\014nition)53 b(1.2.)c Fl(We)g(say)f(that)h(a)f(p)-5 b(erio)g(dic)47 b(p)-5 b(oint)48 b Fr(p)53 b Fq(2)g Fr(M)59 b Fl(of)48 b(p)-5 b(erio)g(d)48 b Fr(l)j Fl(is)56 b Ft(top)s (ologically)43 b(h)m(yp)s(erb)s(olic)-236 3820 y Fl(\()p Fr(p)h Fq(2)g Fr(P)14 b(er)173 3835 y Fm(H)240 3820 y Fl(\))44 b(if)f Fr(f)486 3783 y Fm(l)556 3820 y Fl(is)g(lo)-5 b(c)g(al)5 b(ly)43 b(c)-5 b(onjugate)g(d)43 b(to)h(the)g(line)-5 b(ar)43 b(map)g Fr(L)11 b Ft(:)37 b Fo(R)2478 3777 y Fm(r)2545 3820 y Fq(\002)29 b Fo(R)2723 3777 y Fm(n)p Fk(\000)p Fm(r)2903 3820 y Fq(!)43 b Fo(R)3118 3777 y Fm(r)3185 3820 y Fq(\002)29 b Fo(R)3363 3777 y Fm(n)p Fk(\000)p Fm(r)3543 3820 y Fl(given)42 b(by)-236 3959 y Fr(L)p Ft(\()p Fr(x;)17 b(y)t Ft(\))27 b(=)h(\()p Fr(x=)p Ft(2)p Fr(;)17 b Ft(2)p Fr(y)t Ft(\))p Fl(.)43 b(In)34 b(this)h(c)-5 b(ase)34 b(we)g(say)h(that)h Fr(p)27 b Fq(2)h Fr(P)14 b(er)2013 3923 y Fm(r)2010 3984 y(H)2105 3959 y Fq(\032)28 b Fr(P)14 b(er)2376 3974 y Fm(H)2443 3959 y Fl(,)34 b(we)h(say)g(that)g Fr(r)j Fl(is)c(the)h(index)f(of)h Fr(p)p Fl(.)-137 4198 y Ft(In)24 b(our)f(case)g Fr(f)34 b Ft(is)23 b(expansiv)m(e,)j(so,)f(due)f(to)f(results)g(in)f([)1823 4198 y SDict begin H.S end 1823 4198 a Ft(L1)1933 4130 y SDict begin H.R end 1933 4130 a 1933 4198 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.L1) cvn H.B /ANN pdfmark end 1933 4198 a Ft(])h(\(Lemma)f(2.7\))g(it)g(is)h(true)g(that)g Fr(P)14 b(er)3289 4162 y Fp(0)3286 4224 y Fm(H)3380 4198 y Ft(=)28 b Fr(P)14 b(er)3653 4162 y Fm(n)3650 4224 y(H)3744 4198 y Ft(=)28 b Fq(;)-236 4338 y Ft(since)33 b(no)f(stable)h(p)s(oin)m (ts)f(exist.)-137 4477 y(W)-8 b(e)45 b(denote)g(as)g Fr(H)582 4492 y Fm(k)625 4477 y Ft(\()p Fr(A)p Ft(\))f(\()p 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y(W)-8 b(e)40 b(pro)m(v)m(e)g(a)e(separation)g (prop)s(ert)m(y)i(v)m(eri\014ed)f(b)m(y)h(the)f(stable)g(and)g (unstable)g(set)g(of)f(a)h(p)s(oin)m(t)f Fr(p)g Fq(2)h Fr(P)14 b(er)3807 5418 y Fm(r)3804 5479 y(H)3871 5454 y Ft(.)-236 5593 y(The)35 b(pro)s(of)e(of)h(this)g(Prop)s(osition)e (follo)m(ws)h(the)h(ideas)g(in)g([)1933 5593 y SDict begin H.S end 1933 5593 a Ft(V1)2055 5525 y SDict begin H.R end 2055 5525 a 2055 5593 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.V1) cvn H.B /ANN pdfmark end 2055 5593 a Ft(],[)2136 5593 y SDict begin H.S end 2136 5593 a Ft(V2)2258 5525 y SDict begin H.R end 2258 5525 a 2258 5593 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.V2) cvn H.B /ANN pdfmark end 2258 5593 a Ft(])g(and)g(it)f(is)h(dev)m(elop)s(ed)h (in)e(section)3606 5593 y SDict begin H.S end 3606 5593 a Ft(2)3655 5529 y SDict begin H.R end 3655 5529 a 3655 5593 a SDict 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begin H.R end 2727 1045 a 2727 1113 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.L2) cvn H.B /ANN pdfmark end 2727 1113 a Ft(].)-236 1327 y SDict begin H.S end -236 1327 a -236 1327 a SDict begin 16.79996 H.A end -236 1327 a -236 1327 a SDict begin [/View [/XYZ H.V]/Dest (defi.1.3) cvn /DEST pdfmark end -236 1327 a Fy(De\014nition)52 b(1.3.)d Fl(We)f(say)g(that)h Fr(p)i Fq(2)i Fr(M)58 b Fl(admits)48 b(a)55 b Ft(lo)s(cal)44 b(pro)s(duct)j(structure)i Fl(if)f(ther)-5 b(e)48 b(exists)g(a)f(map)-236 1467 y Fr(h)11 b Ft(:)36 b Fo(R)-35 1425 y Fm(k)35 1467 y Fq(\002)27 b Fo(R)211 1425 y Fm(n)p Fk(\000)p Fm(k)390 1467 y Fq(!)38 b Fr(M)52 b Fl(which)40 b(is)h(a)f(home)-5 b(omorphism)39 b(over)h(its)h(image)f(\()p Fr(p)f Fq(2)g Fr(I)8 b(m)p Ft(\()p Fr(h)p Ft(\))p Fl(\))41 b(and)f(if)h(ther)-5 b(e)41 b(exists)-236 1606 y Fr(")27 b(>)h Ft(0)k Fl(such)h(that)g(for)g (al)5 b(l)32 b Ft(\()p Fr(x;)17 b(y)t Ft(\))27 b Fq(2)h Fo(R)1150 1564 y Fm(k)1210 1606 y Fq(\002)19 b Fo(R)1378 1564 y Fm(n)p Fk(\000)p Fm(k)1551 1606 y Fl(it)33 b(is)f(veri\014e)-5 b(d)32 b(that)i Fr(h)p Ft(\()p Fq(f)p Fr(x)p Fq(g)17 b(\002)i Fo(R)2718 1564 y Fm(n)p Fk(\000)p Fm(k)2858 1606 y Ft(\))28 b(=)f Fr(W)3133 1570 y Fm(s)3119 1631 y(")3170 1606 y Ft(\()p Fr(h)p Ft(\()p Fr(x;)17 b(y)t Ft(\)\))f Fq(\\)i Fr(I)8 b(m)p Ft(\()p Fr(h)p Ft(\))-236 1746 y Fl(and)40 b Fr(h)p Ft(\()p Fo(R)125 1703 y Fm(k)194 1746 y Fq(\002)27 b(f)p Fr(y)t Fq(g)p Ft(\))38 b(=)g Fr(W)746 1710 y Fm(u)732 1770 y(")791 1746 y Ft(\()p Fr(h)p Ft(\()p Fr(x;)17 b(y)t Ft(\)\))25 b Fq(\\)i Fr(I)8 b(m)p Ft(\()p Fr(h)p Ft(\))p Fl(.)63 b(We)41 b(say)g(that)g(the)g(lo)-5 b(c)g(al)40 b(pr)-5 b(o)g(duct)41 b(structur)-5 b(e)42 b(is)e(a)48 b Ft(uniform)-236 1885 y(lo)s(cal)33 b(pro)s(duct)j (structure)j Fl(if)e(in)h(addition)e(to)i(the)g(pr)-5 b(evious)37 b(c)-5 b(onditions,)37 b(ther)-5 b(e)38 b(exists)f Fr(r)e(>)e Ft(0)k Fl(such)h(that)g(for)-236 2025 y(al)5 b(l)34 b Fr(x)28 b Fq(2)g Fr(M)46 b Fl(the)35 b(p)-5 b(oints)34 b(in)h Fr(B)862 2040 y Fm(r)900 2025 y Ft(\()p Fr(x)p Ft(\))g Fl(admit)g(a)f(lo)-5 b(c)g(al)35 b(pr)-5 b(o)g(duct)35 b(structur)-5 b(e.)-137 2239 y Ft(W)d(e)39 b(remark)g(that)f(the)h(p)s(oin)m(ts)g(admitting)c(a)k(lo)s(cal)d(pro)s (duct)j(structure)h(are)e(an)h(op)s(en)g(set.)62 b(W)-8 b(e)39 b(call)e(the)-236 2379 y(p)s(oin)m(ts)32 b(whic)m(h)h(do)g(not)f (admit)f(a)h(lo)s(cal)e(pro)s(duct)j(structure)h(singularities.)-236 2593 y SDict begin H.S end -236 2593 a -236 2593 a SDict begin 16.79996 H.A end -236 2593 a -236 2593 a SDict begin [/View [/XYZ H.V]/Dest (teo.1.1) cvn /DEST pdfmark end -236 2593 a Fy(Theorem)44 b(1.1.)h Fl(L)-5 b(et)41 b Fr(f)21 b Ft(:)36 b Fr(M)49 b Fq(!)37 b Fr(M)52 b Fl(b)-5 b(e)40 b(an)g(exp)-5 b(ansive)39 b(home)-5 b(omorphism)38 b(such)i(that)p 3067 2513 233 4 v 41 w Fr(P)14 b(er)3233 2608 y Fm(H)3338 2593 y Ft(=)38 b Fr(M)10 b Fl(.)63 b(Then,)-236 2733 y(every)36 b(p)-5 b(oint)36 b(in)g Fr(P)14 b(er)558 2748 y Fm(H)661 2733 y Fl(admits)36 b(a)g(lo)-5 b(c)g(al)36 b(pr)-5 b(o)g(duct)36 b(structur)-5 b(e.)50 b(In)36 b(p)-5 b(articular,)36 b(the)h(set)f(of)g(p)-5 b(oints)36 b(with)g(a)g(lo)-5 b(c)g(al)-236 2872 y(pr)g(o)g(duct)35 b(structur)-5 b(e)36 b(is)e(op)-5 b(en)34 b(and)h(dense)f(in)g Fr(M)10 b Fl(.)-137 3086 y Ft(Once)38 b(this)e(is)g(obtained,)h(in)f([)991 3086 y SDict begin H.S end 991 3086 a Ft(V2)1113 3018 y SDict begin H.R end 1113 3018 a 1113 3086 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.V2) cvn H.B /ANN pdfmark end 1113 3086 a Ft(])h(singularities)d(are)j(studied,)h(discarding)d (their)i(existence)h(b)m(y)f(studying)-236 3226 y(the)c(w)m(a)m(y)h(in) e(whic)m(h)i(the)f(pro)s(duct)g(structure)h(is)f(glued)f(together)h(in) f(the)i(singularit)m(y)d(and)i(pro)m(ving)f(that)h(this)-236 3365 y(can)k(not)g(happ)s(en)g(b)m(y)h(discarding)e(the)i(p)s(ossible)e (dimensions)g(in)h(whic)m(h)g(that)g(gluing)e(ma)m(y)i(happ)s(en)g(one) h(b)m(y)-236 3505 y(one)33 b(.)45 b(As)34 b(w)m(as)g(already)f (explained,)g(with)g(the)g(pro)s(duct)h(b)s(et)m(w)m(een)h(the)e(Anoso) m(v)i(and)e(the)g(pseudo-Anso)m(v)j(w)m(e)-236 3644 y(see)e(that)e (this)h(can)g(not)g(b)s(e)g(done)g(in)g(dimension)e(larger)h(than)h(3,) f(unless)i(w)m(e)g(add)f(the)g(h)m(yp)s(othesis)h(of)f(ha)m(ving)-236 3784 y Fr(P)14 b(er)-67 3743 y Fm(n)p Fk(\000)p Fp(1)-70 3811 y Fm(H)97 3784 y Ft(=)27 b Fr(P)14 b(er)366 3799 y Fm(H)433 3784 y Ft(.)44 b(This)32 b(will)f(b)s(e)i(studied)f(in)g (section)1825 3784 y SDict begin H.S end 1825 3784 a Ft(4.3)1950 3720 y SDict begin H.R end 1950 3720 a 1950 3784 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (subsection.4.3) cvn H.B /ANN pdfmark end 1950 3784 a Ft(.)-137 3923 y(It)40 b(is)f(w)m(orth)g(observing)h(that)f(the)h(fact)f(of)f(ha)m(ving)h(a)g (lo)s(cal)f(pro)s(duct)h(structure)i(in)d(an)h(op)s(en)h(and)f(dense) -236 4063 y(subset)24 b(do)s(es)f(not)g(imply)-8 b(,)23 b(a)g(priori,)f(that)h(the)g(index)g(of)g(the)g(top)s(ologically)c(h)m (yp)s(erb)s(olic)j(p)s(erio)s(dic)f(p)s(oin)m(ts)i(should)-236 4202 y(b)s(e)31 b(constan)m(t)g(in)f(all)e(the)k(manifold.)40 b(W)-8 b(e)31 b(shall)e(pro)m(v)m(e)j(this)e(is)g(true,)i(under)f(the)g (h)m(yp)s(othesis)h(of)e(Theorem)3746 4202 y SDict begin H.S end 3746 4202 a Ft(1.1)3871 4138 y SDict begin H.R end 3871 4138 a 3871 4202 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (teo.1.1) cvn H.B /ANN pdfmark end 3871 4202 a Ft(,)-236 4342 y(for)k(dimensions)h(3)f(and)i(4.)51 b(F)-8 b(or)34 b(doing)g(that,)i(in)e(section)1952 4342 y SDict begin H.S end 1952 4342 a Ft(4.1)2077 4278 y SDict begin H.R end 2077 4278 a 2077 4342 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (subsection.4.1) cvn H.B /ANN pdfmark end 2077 4342 a 35 w Ft(sev)m(eral)i(prop)s(erties)f(of)g(the)g(p)s(oin)m (ts)g(in)p 3592 4246 306 4 v 34 w Fr(P)14 b(er)3761 4301 y Fm(n)p Fk(\000)p Fp(1)3758 4369 y Fm(H)-236 4481 y Ft(are)32 b(studied.)44 b(The)34 b(follo)m(wing)c(sharp)s(er)j(result)f (is)g(obtained.)-236 4696 y SDict begin H.S end -236 4696 a -236 4696 a SDict begin 16.79996 H.A end -236 4696 a -236 4696 a SDict begin [/View [/XYZ H.V]/Dest (teo.1.2) cvn /DEST pdfmark end -236 4696 a Fy(Theorem)44 b(1.2.)g Fl(L)-5 b(et)41 b Fr(f)48 b Ft(:)38 b Fr(M)49 b Fq(!)37 b Fr(M)51 b Fl(b)-5 b(e)40 b(an)g(exp)-5 b(ansive)38 b(home)-5 b(omorphism)38 b(verifying)p 3069 4616 233 4 v 40 w Fr(P)14 b(er)3235 4711 y Fm(H)3339 4696 y Ft(=)38 b Fr(M)10 b Fl(.)62 b(Then,)-236 4835 y Fr(P)14 b(er)-67 4794 y Fm(n)p Fk(\000)p Fp(1)-70 4862 y Fm(H)97 4835 y Ft(=)27 b Fr(P)14 b(er)366 4850 y Fm(H)468 4835 y Fl(or)35 b Fr(P)14 b(er)763 4794 y Fm(n)p Fk(\000)p Fp(1)760 4862 y Fm(H)927 4835 y Ft(=)28 b Fq(;)p Fl(.)44 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b Fr(P)14 b(er)1239 3635 y Fm(k)1236 3696 y(H)1337 3671 y Fl(is)35 b(op)-5 b(en)34 b(in)h Fr(P)14 b(er)1957 3686 y Fm(H)2058 3671 y Fl(and)-236 3696 y SDict begin H.S end -236 3696 a -236 3696 a SDict begin 16.79996 H.A end -236 3696 a -236 3696 a SDict begin [/View [/XYZ H.V]/Dest (Item.2) cvn /DEST pdfmark end -236 3696 a -88 3810 a Ft(\(2\))41 b Fl(for)26 b(al)5 b(l)26 b Fr(p)i Fq(2)g Fr(P)14 b(er)693 3825 y Fm(H)786 3810 y Fl(ther)-5 b(e)26 b(exists)g(op)-5 b(en)25 b(neighb)-5 b(orho)g(o)g(ds)24 b(of)i Fr(p)p Fl(,)i Fr(V)2378 3825 y Fp(1)2443 3810 y Fl(and)e Fr(V)2681 3825 y Fp(2)2746 3810 y Fl(such)g(that)g(for)g(al)5 b(l)26 b Fr(q)32 b Fq(2)c Fr(P)14 b(er)3762 3825 y Fm(H)3831 3810 y Fq(\\)78 3950 y Fr(V)135 3965 y Fp(1)202 3950 y Fl(we)27 b(have)g Fr(S)616 3965 y Fm(q)660 3950 y Fq(\\)6 b Fr(U)798 3965 y Fm(p)866 3950 y Fq(6)p Ft(=)28 b Fq(;)f Fl(and)g Fr(U)1295 3965 y Fm(q)1340 3950 y Fq(\\)6 b Fr(S)1472 3965 y Fm(p)1539 3950 y Fq(6)p Ft(=)28 b Fq(;)p Fl(,)h(wher)-5 b(e)27 b 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Fp(1)-119 4592 y Fq(2)28 b Ft(\(0)p Fr(;)17 b(\032)p Ft(\))32 b(and)h(let)f Fr(V)614 4607 y Fp(2)681 4592 y Ft(=)c Fr(h)p Ft(\()p Fr(B)953 4607 y Fm(\032)989 4616 y Fg(1)1028 4592 y Ft(\(0\)\).)43 b(F)-8 b(or)31 b Fr(q)h Fq(2)c Fr(h)p Ft(\()p Fr(B)1772 4607 y Fm(\032)1808 4616 y Fg(1)1847 4592 y Ft(\(0\)\))k(w)m(e)i(denote)f Fr(S)2566 4556 y Fk(0)2560 4617 y Fm(q)2626 4592 y Ft(=)27 b Fr(h)2785 4556 y Fk(\000)p Fp(1)2880 4592 y Ft(\()p Fr(S)2978 4607 y Fm(q)3016 4592 y Ft(\))32 b(and)h Fr(U)3352 4556 y Fk(0)3342 4617 y Fm(q)3408 4592 y Ft(=)28 b Fr(h)3568 4556 y Fk(\000)p Fp(1)3662 4592 y Ft(\()p Fr(U)3766 4607 y Fm(q)3804 4592 y Ft(\).)-137 4732 y(Let)j Fr(\032)86 4747 y Fp(2)157 4732 y Ft(and)f Fr(\032)394 4747 y Fp(3)465 4732 y Ft(giv)m(en)g(b)m(y)i(Lemmas)1235 4732 y SDict begin H.S end 1235 4732 a Ft(3.2)1360 4668 y SDict begin H.R end 1360 4668 a 1360 4732 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lema.3.2) cvn H.B /ANN pdfmark end 1360 4732 a 31 w Ft(and)1578 4732 y SDict begin H.S end 1578 4732 a Ft(3.3)1703 4668 y SDict begin H.R end 1703 4668 a 1703 4732 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lema.3.3) cvn H.B /ANN pdfmark end 1703 4732 a 31 w Ft(suc)m(h)g(that)e(if) g(dist)o(\()p Fr(h)2500 4696 y Fk(\000)p Fp(1)2595 4732 y Ft(\()p Fr(q)t Ft(\))p Fr(;)17 b Ft(0\))26 b Fr(<)i(\032)3029 4747 y Fp(3)3099 4732 y Ft(then)k Fr(U)3396 4696 y Fk(0)3386 4756 y Fm(q)3442 4732 y Fq(\\)18 b Fr(B)3600 4747 y Fm(\032)3636 4756 y Fg(2)3675 4732 y Ft(\()p 3713 4652 100 4 v Fr(S)3779 4703 y Fk(0)3773 4756 y Fm(p)3831 4732 y Fq(\\)-236 4871 y Fr(@)5 b(B)-106 4886 y Fm(\032)-70 4895 y Fg(1)-30 4871 y Ft(\(0\)\))30 b(=)i Fq(;)i Ft(and)h Fr(S)613 4835 y Fk(0)607 4896 y Fm(q)676 4871 y Fq(\032)d Fr(B)859 4886 y Fm(\032)895 4895 y Fg(2)934 4871 y Ft(\()p Fr(S)1038 4835 y Fk(0)1032 4896 y Fm(p)1071 4871 y Ft(\).)50 b(Let)35 b Fr(V)1420 4886 y Fp(1)1490 4871 y Ft(=)c Fr(h)p Ft(\()p Fr(B)1765 4886 y Fm(\032)1801 4895 y Fg(3)1841 4871 y Ft(\(0\)\).)49 b(Observ)m(e)36 b(that)f(w)m(e)h(can)f(use)g(Lemma)3517 4871 y SDict begin H.S end 3517 4871 a Ft(3.2)3642 4807 y SDict begin H.R end 3642 4807 a 3642 4871 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lema.3.2) cvn H.B /ANN pdfmark end 3642 4871 a 35 w Ft(to)f Fr(S)3858 4886 y Fm(p)-236 5011 y Ft(and)e Fr(U)19 5026 y Fm(p)92 5011 y Ft(b)s(ecause)i(of)e(the) h(c)m(hoice)g(of)f Fr(\032)1183 5026 y Fp(1)1223 5011 y Ft(.)-137 5150 y(By)41 b(applying)f(Lemma)787 5150 y SDict begin H.S end 787 5150 a Ft(3.1)911 5086 y SDict begin H.R end 911 5086 a 911 5150 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lema.3.1) cvn H.B /ANN pdfmark end 911 5150 a 41 w Ft(w)m(e)h(kno)m(w)h(that)e(if)g Fr(q)45 b Fq(2)c Fr(V)1935 5165 y Fp(1)2002 5150 y Fq(\\)28 b Fr(P)14 b(er)2265 5114 y Fm(m)2262 5175 y(H)2371 5150 y Ft(then)41 b(there)h(exists)f Fr(h)28 b Fq(\016)f Fr(\015)46 b Fq(\032)c Fr(S)3573 5165 y Fm(q)3651 5150 y Ft(a)e(non)-236 5290 y(trivial)28 b(cycle)j(of)f(the)h Fr(m)19 b Fq(\000)g Ft(1)30 b(dimensional)e(homology)h(of)h Fr(V)1986 5305 y Fp(2)2044 5290 y Fq(n)18 b Fr(U)2178 5305 y Fm(q)2216 5290 y Ft(.)43 b(Because)33 b(of)d(Lemma)3113 5290 y SDict begin H.S end 3113 5290 a Ft(3.1)3238 5225 y SDict begin H.R end 3238 5225 a 3238 5290 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lema.3.1) cvn H.B /ANN pdfmark end 3238 5290 a 30 w Ft(as)h(w)m(ell,)g(w)m(e)g(can)-236 5429 y(supp)s(ose)j(that)e Fr(\015)h Fq(\032)28 b Fr(B)605 5444 y Fm(\032)641 5453 y Fg(2)680 5429 y Ft(\()p 718 5349 100 4 v Fr(S)784 5400 y Fk(0)778 5454 y Fm(p)840 5429 y Fq(\\)22 b Fr(@)5 b(B)1058 5444 y Fm(\032)1094 5453 y Fg(1)1134 5429 y Ft(\(0\)\).)-137 5569 y(Let)35 b Fr(\031)95 5584 y Fm(t)136 5569 y Ft(:)f Fo(R)269 5526 y Fm(k)335 5569 y 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/I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (teo.1.1) cvn H.B /ANN pdfmark end 895 1221 a Fs(.)-137 1360 y Ft(W)-8 b(e)34 b(are)f(going)f(to)h(construct)i(a)e(lo)s(cal)e(pro)s(duct)i (structure)i(in)d(a)h(neigh)m(b)s(orho)s(o)s(d)g(of)f(ev)m(ery)j Fr(p)29 b Fq(2)h Fr(P)14 b(er)3623 1375 y Fm(H)3690 1360 y Ft(.)45 b(W)-8 b(e)-236 1500 y(consider)33 b(the)g(notation)e(of)h (the)h(statemen)m(t)g(of)f(Prop)s(osition)2070 1500 y SDict begin H.S end 2070 1500 a Ft(3.1)2195 1436 y SDict begin H.R end 2195 1436 a 2195 1500 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (prop.3.1) cvn H.B /ANN pdfmark end 2195 1500 a Ft(.)-137 1639 y(Let)e Fr(\031)90 1654 y Fm(s)138 1639 y Ft(:)p 199 1559 97 4 v 34 w Fr(V)256 1654 y Fp(1)322 1639 y Fq(!)e Fr(S)510 1654 y Fm(p)579 1639 y Ft(b)s(e)h(de\014ned)i(in)e(the)h(p)s(oin)m(ts)f Fr(q)i Fq(2)d Fr(P)14 b(er)1941 1654 y Fm(H)2038 1639 y Ft(as)29 b Fr(\031)2209 1654 y Fm(s)2246 1639 y Ft(\()p Fr(q)t Ft(\))f(=)f Fr(U)2566 1654 y Fm(q)2620 1639 y Fq(\\)16 b Fr(S)2762 1654 y Fm(p)2802 1639 y Ft(.)42 b(This)30 b(map)f(is)f(w)m(ell)h(de\014ned)-236 1779 y(in)40 b(a)i(dense)h(subset)g(of)e Fr(V)737 1794 y Fp(1)817 1779 y Ft(b)s(ecause)i(of)e(Prop)s(osition)1841 1779 y SDict begin H.S end 1841 1779 a Ft(3.1)1966 1715 y SDict begin H.R end 1966 1715 a 1966 1779 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (prop.3.1) cvn H.B /ANN pdfmark end 1966 1779 a Ft(.)70 b(Let)41 b Fr(x)i Fq(2)p 2454 1699 97 4 v 44 w Fr(V)2511 1794 y Fp(1)2591 1779 y Ft(and)f Fr(q)2833 1794 y Fm(n)2923 1779 y Fq(!)g Fr(x)p Ft(,)j Fr(q)3235 1794 y Fm(n)3325 1779 y Fq(2)e Fr(P)14 b(er)3600 1794 y Fm(H)3708 1779 y Ft(with)-236 1918 y Fr(\031)t Ft(\()p Fr(q)-96 1933 y Fm(n)-49 1918 y Ft(\))39 b Fq(!)g Fr(y)t Ft(.)64 b(Observ)m(e)41 b(that)e Fr(y)k Fq(2)d Fr(W)1212 1882 y Fm(u)1198 1943 y(")1257 1918 y Ft(\()p Fr(x)p Ft(\))27 b Fq(\\)g Fr(S)1568 1933 y Fm(p)1647 1918 y Ft(and)40 b(expansivit)m(y)g(imply)e(that)h(the)h(in)m (tersection)f(p)s(oin)m(t)g(is)-236 2058 y(unique.)44 b(This)32 b(allo)m(ws)g(us)h(to)f(extend)i Fr(\031)1248 2073 y Fm(s)1318 2058 y Ft(to)p 1437 1978 V 32 w Fr(V)1494 2073 y Fp(1)1533 2058 y Ft(.)44 b(The)33 b(same)g(reason)g(ensures)h (this)e(extension)i(is)e(con)m(tin)m(uous.)-236 2197 y(Also)23 b(w)m(e)i(ha)m(v)m(e)g Fr(\031)381 2212 y Fm(s)418 2197 y Ft(\()p Fr(x)p Ft(\))j Fq(2)g Fr(U)737 2212 y Fm(x)785 2197 y Fq(\\)t Fr(S)915 2212 y Fm(p)979 2197 y Ft(and)c(b)s(ecause)h(of)e(expansivit)m(y)i Fr(\031)2176 2212 y Fm(s)2213 2197 y Ft(\()p Fr(x)p Ft(\))j(=)g Fr(U)2542 2212 y Fm(x)2590 2197 y Fq(\\)t Fr(S)2720 2212 y Fm(p)2784 2197 y Ft(for)23 b(all)f Fr(x)28 b Fq(2)g Fr(V)3285 2212 y Fp(1)3324 2197 y Ft(.)41 b(Expansivit)m(y)-236 2337 y(also)31 b(implies)g(that)h Fr(\031)557 2352 y Fm(s)594 2337 y Fq(j)p Fr(S)682 2352 y Fm(x)758 2337 y Ft(is)g(injectiv)m(e.) -137 2476 y(If)e Fr(q)i Fq(2)c Fr(P)14 b(er)293 2491 y Fm(H)390 2476 y Ft(then)31 b(the)f(In)m(v)-5 b(ariance)30 b(of)g(Domain)e(Theorem)i(\(see)h([)2344 2496 y SDict begin H.S end 2344 2496 a -20 x Ft(Sp)2453 2408 y SDict begin H.R end 2453 2408 a 2453 2476 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Sp) cvn H.B /ANN pdfmark end 2453 2476 a Ft(]\))f(implies)d(that)j Fr(\031)3140 2491 y Fm(s)3177 2476 y Fq(j)p Fr(S)3265 2491 y Fm(q)3333 2476 y Ft(is)f(op)s(en)i(and)f(a)-236 2616 y(homeomorphism)24 b(o)m(v)m(er)k(its)e(image.)40 b(Observ)m(e)29 b(that)e Fr(\031)1771 2631 y Fm(s)1808 2616 y Ft(\()p Fr(r)s Ft(\))g Fq(2)h Fr(\031)2107 2631 y Fm(s)2144 2616 y Ft(\()p Fr(S)2242 2631 y Fm(q)2280 2616 y Ft(\))f(with)f Fr(r)m(;)17 b(q)32 b Fq(2)c Fr(P)14 b(er)2981 2631 y Fm(H)3075 2616 y Ft(implies)24 b Fr(U)3466 2631 y Fm(r)3515 2616 y Fq(\\)10 b Fr(S)3651 2631 y Fm(q)3717 2616 y Fq(6)p Ft(=)28 b Fq(;)p Ft(.)-236 2760 y(Let)33 b Fr(W)41 b Fq(\032)28 b Fr(S)237 2775 y Fm(p)277 2760 y Ft(,)k Fr(p)c Fq(2)g Fr(W)14 b Ft(,)32 b Fr(W)46 b Ft(homeomorphic)31 b(to)h(the)h(disk)p 1957 2680 84 4 v 33 w Fr(D)2041 2698 y Fm(k)2116 2760 y Ft(and)g Fr(W)46 b Ft(relativ)m(e)32 b(neigh)m(b)s(orho)s(o)s(d)f(of)h Fr(p)h Ft(in)f Fr(S)3770 2775 y Fm(p)3809 2760 y Ft(.)-137 3039 y(W)-8 b(e)42 b(a\016rm)e(there)h(exists)h Fr(V)922 3054 y Fp(3)1002 3039 y Ft(neigh)m(b)s(orho)s(o)s(d)d(of)i Fr(p)81 b Ft(suc)m(h)42 b(that)f(for)f(all)f Fr(q)45 b Fq(2)d Fr(V)2874 3054 y Fp(3)2941 3039 y Fq(\\)28 b Fr(P)14 b(er)3201 3054 y Fm(H)3268 3039 y Ft(,)43 b Fr(W)55 b Fq(\032)43 b Fr(\031)3660 3054 y Fm(s)3697 3039 y Ft(\()p Fr(S)3795 3054 y Fm(q)3833 3039 y Ft(\).)-236 3178 y(Otherwise,)c Fr(q)301 3193 y Fm(n)383 3178 y Fq(!)c Fr(p)i Ft(w)m(ould)g(exists,)i (suc)m(h)g(that)d Fr(q)1678 3193 y Fm(n)1761 3178 y Fq(2)g Fr(P)14 b(er)2029 3193 y Fm(H)2133 3178 y Ft(and)37 b Fr(W)49 b Fo(*)35 b Fr(\031)2635 3193 y Fm(s)2673 3178 y Ft(\()p Fr(S)2771 3193 y Fm(q)2803 3201 y Fi(n)2849 3178 y Ft(\).)57 b(Since)37 b Fr(W)50 b Ft(is)37 b(connected)-236 3318 y(and)29 b Fr(\031)5 3333 y Fm(s)42 3318 y Fq(j)p Fr(S)130 3333 y Fm(q)162 3341 y Fi(n)237 3318 y Ft(op)s(en,)i Fr(y)546 3333 y Fm(n)620 3318 y Fq(2)d Fr(@)5 b(\031)825 3333 y Fm(s)863 3318 y Ft(\()p Fr(S)961 3333 y Fm(q)993 3341 y Fi(n)1039 3318 y Ft(\))15 b Fq(\\)g Fr(W)44 b Ft(m)m(ust)30 b(exist)f(\(the)h(fron)m(tier)e(is)h(relativ)m(e)g(to)g Fr(S)2936 3333 y Fm(p)2975 3318 y Ft(\).)43 b(So)29 b(there)h(m)m(ust)f (exist)-236 3457 y Fr(x)-181 3472 y Fm(n)-106 3457 y Fq(2)f Fr(@)5 b(V)101 3472 y Fp(1)153 3457 y Fq(\\)12 b Fr(S)291 3472 y Fm(q)323 3480 y Fi(n)398 3457 y Ft(suc)m(h)29 b(that)e Fr(\031)874 3472 y Fm(s)911 3457 y Ft(\()p Fr(x)1004 3472 y Fm(n)1052 3457 y Ft(\))g(=)h Fr(y)1269 3472 y Fm(n)1315 3457 y Ft(.)42 b(W)-8 b(e)28 b(can)g(supp)s(ose)h Fr(x)2138 3472 y Fm(n)2213 3457 y Fq(!)e Fr(x)h Ft(and)g Fr(y)2656 3472 y Fm(n)2730 3457 y Fq(!)g Fr(y)i Ft(p)s(oin)m(ts)e(of)f Fr(S)3391 3472 y Fm(q)3423 3480 y Fi(n)3481 3457 y Fq(\\)12 b Fr(@)5 b(V)3672 3472 y Fp(1)3740 3457 y Ft(and)-236 3597 y Fr(W)52 b Ft(resp)s(ectiv)m(ely)-8 b(,)40 b(the)f(\014rst)g(due) f(to)g(semicon)m(tin)m(uit)m(y)g(of)g(lo)s(cal)d(stable)j(sets)i (\(Lemma)3046 3597 y SDict begin H.S end 3046 3597 a Ft(3.2)3170 3532 y SDict begin H.R end 3170 3532 a 3170 3597 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lema.3.2) cvn H.B /ANN pdfmark end 3170 3597 a Ft(\))e(and)h(the)f(second)-236 3736 y(b)s(ecause)31 b Fr(W)44 b Ft(is)30 b(compact.)42 b(F)-8 b(rom)29 b(the)i(construction)f(of)g Fr(\031)1921 3751 y Fm(s)1988 3736 y Ft(w)m(e)h(deduce)h(that)e Fr(x)g Ft(and)h Fr(y)i Ft(are)d(o)m(v)m(er)i(the)e(same)g(lo-)-236 3876 y(cal)25 b(stable)g(and)h(unstable)g(set)g(con)m(tradicting)f (expansivit)m(y)h(\(observ)m(e)i(that)d(dist\()p Fr(W)14 b Ft(,)p Fr(S)2963 3891 y Fm(p)3011 3876 y Fq(\\)8 b Fr(@)d(V)3198 3891 y Fp(1)3238 3876 y Ft(\))28 b Fr(>)f Ft(0)f(so)g Fr(x)i Fq(6)p Ft(=)f Fr(y)t Ft(\).)-137 4155 y(Let)33 b Fr(V)95 4170 y Fp(4)162 4155 y Ft(=)27 b Fr(\031)324 4118 y Fk(\000)p Fp(1)320 4179 y Fm(s)418 4155 y Ft(\()p Fr(W)14 b Ft(\))21 b Fq(\\)h Fr(V)766 4170 y Fp(3)805 4155 y Ft(,)33 b(w)m(e)g(ha)m(v)m(e)g(that)f(for)g(ev)m(ery)i Fr(q)t(;)17 b(r)30 b Fq(2)e Fr(P)14 b(er)2274 4170 y Fm(H)2362 4155 y Fq(\\)21 b Fr(V)2506 4170 y Fp(4)2546 4155 y Ft(,)32 b(it)f(is)h(true)g(that)g Fr(S)3276 4170 y Fm(q)3335 4155 y Fq(\\)22 b Fr(U)3489 4170 y Fm(r)3555 4155 y Fq(6)p Ft(=)28 b Fq(;)j Ft(and)-236 4294 y Fr(S)-176 4309 y Fm(r)-116 4294 y Fq(\\)22 b Fr(U)38 4309 y Fm(q)104 4294 y Fq(6)p Ft(=)28 b Fq(;)k Ft(from)g(its)g(construction.)-137 4433 y(Let)c Fr(A)106 4448 y Fm(s)171 4433 y Fq(\032)g Fr(S)336 4448 y Fm(p)387 4433 y Fq(\\)12 b Fr(V)522 4448 y Fp(4)589 4433 y Ft(and)28 b Fr(B)848 4448 y Fm(u)920 4433 y Fq(\032)h Fr(U)1092 4448 y Fm(p)1143 4433 y Fq(\\)12 b Fr(V)1278 4448 y Fp(4)1345 4433 y Ft(b)s(e)28 b(relativ)m(e)f(neigh)m (b)s(orho)s(o)s(ds)f(of)h Fr(p)p Ft(,)i(b)s(oth)e(homeomorphic)f(to)h (disks.)-236 4573 y(No)m(w,)38 b(let)d Fr(x)g Fq(2)f Fr(A)425 4588 y Fm(s)498 4573 y Ft(and)j Fr(y)g Fq(2)d Fr(B)951 4588 y Fm(u)996 4573 y Ft(,)k(then,)f(b)m(y)h(taking)d(limit)e (p)s(oin)m(ts)j(of)g(the)g(in)m(tersection)h(of)e(lo)s(cal)f(stable)i (and)-236 4712 y(unstable)e(sets)h(of)e(p)s(erio)s(dic)f(p)s(oin)m(ts)h (con)m(v)m(erging)h(to)g Fr(x)g Ft(and)f Fr(y)k Ft(resp)s(ectiv)m(ely) -8 b(,)35 b(semicon)m(tin)m(uit)m(y)e(of)g(lo)s(cal)f(stable)-236 4852 y(and)i(unstable)h(sets)h(\(Lemma)929 4852 y SDict begin H.S end 929 4852 a Ft(3.2)1054 4788 y SDict begin H.R end 1054 4788 a 1054 4852 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lema.3.2) cvn H.B /ANN pdfmark end 1054 4852 a Ft(\))e(and)h(expansivit)m(y)g(easily)f(imply)f(that)h Fr(U)2661 4867 y Fm(x)2729 4852 y Fq(\\)23 b Fr(S)2878 4867 y Fm(y)2954 4852 y Ft(is)34 b(a)h(unique)g(p)s(oin)m(t.)48 b(Let)-236 4991 y Fr(h)11 b Ft(:)33 b Fr(A)-36 5006 y Fm(s)22 4991 y Fq(\002)22 b Fr(B)195 5006 y Fm(u)267 4991 y Fq(!)28 b Fr(V)452 5006 y Fp(1)523 4991 y Ft(giv)m(en)k(b)m(y)h Fr(h)p Ft(\()p Fr(x;)17 b(y)t Ft(\))27 b(=)h Fr(U)1392 5006 y Fm(x)1457 4991 y Fq(\\)21 b Fr(S)1604 5006 y Fm(y)1645 4991 y Ft(.)44 b(It)32 b(is)f(con)m(tin)m(uous)i(and)f(injectiv)m(e.)43 b(Using)32 b(the)g(In)m(v)-5 b(ariance)-236 5131 y(of)33 b(Domain)d(Theorem)k(again)e(w)m(e)i(conclude)g(that)f(it)f(is)g(op)s (en.)46 b(This)33 b(concludes)i(the)e(pro)s(of)g(of)f(the)i(existence) -236 5270 y(of)e(a)g(lo)s(cal)e(pro)s(duct)j(structure)h(in)e(an)g(op)s (en)h(and)g(dense)h(set.)3820 5410 y Fj(\003)-236 5712 y SDict begin H.S end -236 5712 a -236 5712 a SDict begin 16.79996 H.A end -236 5712 a -236 5712 a SDict begin [/View [/XYZ H.V]/Dest (obs.3.1) cvn /DEST pdfmark end -236 5712 a Fl(R)-5 b(emark)35 b Ft(3.1)p Fl(.)f Ft(Although)21 b(this)h(do)s(es)h(not)g(ensure)g(the)g(dimension)e(of)h(the)h(decomp)s 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Fr(P)14 b(er)3646 822 y Fm(n)p Fk(\000)p Fp(1)3643 890 y Fm(H)3816 863 y Ft(in)-236 1003 y(a)33 b(neigh)m(b)s(orho)s(o)s(d)g(of)g(a)h(singularit)m(y)e Fr(x)e Fq(2)g Fr(M)45 b Ft(de\014ning)34 b(a)f(partial)f(order)i(in)f Fr(P)14 b(er)2767 962 y Fm(n)p Fk(\000)p Fp(1)2764 1030 y Fm(H)2903 1003 y Ft(.)48 b(W)-8 b(e)34 b(consider)g Fr(B)3602 1018 y Fm(\027)3645 1003 y Ft(\()p Fr(x)p Ft(\))g(so)-236 1142 y(that)e(Prop)s(osition)500 1142 y SDict begin H.S end 500 1142 a Ft(1.1)625 1078 y SDict begin H.R end 625 1078 a 625 1142 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (prop.1.1) cvn H.B /ANN pdfmark end 625 1142 a 32 w Ft(holds.)43 b(Let)1289 1438 y Fr(S)1349 1453 y Fm(p)1416 1438 y Ft(=)28 b Fr(cc)1604 1453 y Fm(p)1643 1438 y Ft(\()p Fr(W)1787 1396 y Fm(s)1824 1438 y Ft(\()p Fr(p)p Ft(\))22 b Fq(\\)g Fr(B)2133 1453 y Fm(\027)2176 1438 y Ft(\()p Fr(x)p Ft(\)\))p Fr(;)1281 1634 y(U)1347 1649 y Fm(p)1415 1634 y Ft(=)28 b Fr(cc)1603 1649 y Fm(p)1642 1634 y Ft(\()p Fr(W)1786 1593 y Fm(u)1831 1634 y Ft(\()p Fr(p)p Ft(\))22 b Fq(\\)g Fr(B)2140 1649 y Fm(\027)2184 1634 y Ft(\()p Fr(x)p Ft(\)\))p Fr(:)-137 1814 y Ft(Also,)42 b(w)m(e)f(shall)e(supp)s(ose)j(that,)g(b)s (ecause)f(of)f(Remark)1992 1814 y SDict begin H.S end 1992 1814 a Ft(1.1)2116 1750 y SDict begin H.R end 2116 1750 a 2116 1814 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (obs.1.1) cvn H.B /ANN pdfmark end 2116 1814 a Ft(,)i Fr(S)2245 1829 y Fm(p)2326 1814 y Fq(\032)f Fr(W)2550 1778 y Fm(s)2536 1839 y(")2586 1814 y Ft(\()p Fr(p)p Ft(\))f(and)h Fr(U)3015 1829 y Fm(p)3095 1814 y Fq(\032)g Fr(W)3319 1778 y Fm(u)3305 1839 y(")3364 1814 y Ft(\()p Fr(p)p Ft(\))f(for)g(some)-236 1954 y Fr(")27 b(>)h Ft(0.)43 b(F)-8 b(or)32 b(ev)m(ery)i Fr(p)28 b Fq(2)g Fr(P)14 b(er)832 1912 y Fm(n)p Fk(\000)p Fp(1)829 1981 y Fm(H)990 1954 y Fq(\\)23 b Fr(B)1153 1969 y Fm(\027)1196 1954 y Ft(\()p Fr(x)p Ft(\))33 b(w)m(e)h(de\014ne) 42 b(^)-58 b Fr(p)28 b Ft(=)g Fr(B)2040 1969 y Fm(\027)2105 1954 y Fq(n)22 b Fr(cc)2261 1969 y Fm(x)2305 1954 y Ft(\()p Fr(B)2417 1969 y Fm(\027)2460 1954 y Ft(\()p Fr(x)p Ft(\))h Fq(n)e Fr(S)2745 1969 y Fm(p)2785 1954 y Ft(\).)-137 2093 y(Giv)m(en)32 b Fr(\016)g(>)c Ft(0)j(w)m(e)i(de\014ne)h(the)e (follo)m(wing)d(order)k(relation)d(in)h Fr(X)2213 2108 y Fm(\016)2278 2093 y Ft(=)d Fr(P)14 b(er)2551 2052 y Fm(n)p Fk(\000)p Fp(1)2548 2120 y Fm(H)2709 2093 y Fq(\\)21 b Fr(B)2870 2108 y Fm(\016)2908 2093 y Ft(\()p Fr(x)p Ft(\).)44 b(If)31 b Fr(p;)17 b(q)32 b Fq(2)c Fr(X)3549 2108 y Fm(\016)3619 2093 y Ft(w)m(e)33 b(sa)m(y)-236 2233 y(that)e Fr(p)c Fq(\024)h Fr(q)35 b Ft(if)j(^)-57 b Fr(p)28 b Fq(\032)35 b Ft(^)-56 b Fr(q)t Ft(.)43 b(Clearly)30 b(this)h(is)f(a)h(partial)e(order)i(whic)m(h)g(dep)s(ends)i(on)e(the)g (singularit)m(y)f Fr(x)e Fq(2)g Fr(M)10 b Ft(,)32 b Fr(\027)i(>)28 b Ft(0)-236 2372 y(from)j(Prop)s(osition)518 2372 y SDict begin H.S end 518 2372 a Ft(1.1)643 2308 y SDict begin H.R end 643 2308 a 643 2372 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (prop.1.1) cvn H.B /ANN pdfmark end 643 2372 a 32 w Ft(and)h Fr(\016)g Fq(2)c Ft(\(0)p Fr(;)17 b(\027)6 b Ft(\).)43 b(W)-8 b(e)33 b(call)d(c)m(hain)i(to)g(ev)m(ery)i(totally)c (ordered)j(subset)h(of)e(the)g(relation.)-236 2591 y SDict begin H.S end -236 2591 a -236 2591 a SDict begin 16.79996 H.A end -236 2591 a -236 2591 a SDict begin [/View [/XYZ H.V]/Dest (obs.4.1) cvn /DEST pdfmark end -236 2591 a Fl(R)-5 b(emark)48 b Ft(4.1)p Fl(.)43 b Ft(Since)37 b(stable)f(sets)i(of)e(di\013eren)m(t)h(p)s(erio)s(dic)e(p)s(oin)m(ts)h (ha)m(v)m(e)i(empt)m(y)f(in)m(tersection,)g(w)m(e)h(ha)m(v)m(e)g(that) -236 2730 y(if)i(^)-57 b Fr(p)22 b Fq(\\)30 b Ft(^)-56 b Fr(q)32 b Fq(6)p Ft(=)c Fq(;)33 b Ft(then)g(the)g(p)s(oin)m(ts)g Fr(p)g Ft(and)g Fr(q)j Ft(m)m(ust)e(b)s(e)f(related)f(b)m(y)i(the)f (ordering.)44 b(So,)33 b(if)f Fr(p)c Fq(\024)h Fr(q)36 b Ft(and)d Fr(p)c Fq(\024)f Fr(r)36 b Ft(then)-228 2870 y(^)-57 b Fr(p)27 b Fq(\032)36 b Ft(^)-56 b Fr(q)19 b Fq(\\)h Ft(^)-54 b Fr(r)s Ft(,)30 b Fr(q)j Ft(and)c Fr(r)j Ft(m)m(ust)e(b)s(e)f(related.)42 b(This)30 b(implies)c(that)j(if)g Fr(p)e Fq(\024)h Fr(q)33 b Ft(and)d Fq(C)35 b Ft(is)29 b(a)g(maximal)d(c)m(hain)j(con)m(taining)-236 3009 y Fr(p)p Ft(,)j(then)i Fr(q)d Fq(2)d(C)6 b Ft(.)-137 3228 y(This)35 b(order)f(can)g(b)s(e)h(w)m(ell)e(understo)s(o)s(d)h(in)g (the)g(case)h(of)f(surfaces)h(where,)h(for)e(the)g(pseudo)i(Anoso)m(v)f (maps,)-236 3367 y(singularities)30 b(ha)m(v)m(e)k(more)e(than)g(2)g (maximal)e(c)m(hains.)-236 3586 y SDict begin H.S end -236 3586 a -236 3586 a SDict begin 16.79996 H.A end -236 3586 a -236 3586 a SDict begin [/View [/XYZ H.V]/Dest (lema.4.1) cvn /DEST pdfmark end -236 3586 a Fy(Lemma)37 b(4.1.)42 b Fl(Given)35 b(a)g(singularity)g Fr(x)28 b Fq(2)h Fr(M)46 b Fl(and)34 b Fr(\027)h(>)27 b Ft(0)35 b Fl(ther)-5 b(e)35 b(exists)g Fr(\016)d(>)c Ft(0)35 b Fl(such)f(that)i(ther)-5 b(e)35 b(ar)-5 b(e)35 b(\014nitely)-236 3725 y(many)j(maximal)f(chains) g(in)h Fr(X)942 3740 y Fm(\016)980 3725 y Fl(.)55 b(These)37 b(ar)-5 b(e)38 b(p)-5 b(airwise)38 b(disjoint)f(and)h(every)g(one)g(of) g(them)g(ac)-5 b(cumulates)38 b(in)-236 3865 y Fr(x)p Fl(.)-236 4113 y Fs(Pr)n(oof.)124 b Ft(Let)38 b(us)i(supp)s(ose)g (there)f(w)m(ere)h(in\014nitely)d(man)m(y)i(maximal)c(c)m(hains)k (di\013eren)m(t)g(from)f(eac)m(h)h(other.)-236 4253 y(W)-8 b(e)32 b(shall)f(pro)m(v)m(e)j(this)e(implies)d(the)k(existence)h(of)e (arbitrarily)d(large)i(sets)j(of)d(p)s(oin)m(ts)h(whic)m(h)h(are)f(not) g(pairwise)-236 4392 y(related)g(b)m(y)h(the)g(order)g(relation.)42 b(W)-8 b(e)33 b(pro)m(v)m(e)h(this)e(using)g(induction.)-137 4532 y(Let)g Fr(p)86 4547 y Fp(1)125 4532 y Fr(;)17 b(:::;)g(p)343 4547 y Fm(l)396 4532 y Fq(2)28 b Fr(X)571 4547 y Fm(\016)640 4532 y Ft(b)s(e)k(pairwise)e(not)h(related.)42 b(Let)32 b Fq(C)1910 4547 y Fm(i)1969 4532 y Ft(b)s(e)g(maximal)27 b(c)m(hains)32 b(suc)m(h)g(that)f Fr(p)3269 4547 y Fm(i)3325 4532 y Fq(2)d(C)3471 4547 y Fm(i)3531 4532 y Ft(and)j(tak)m(e)-236 4671 y Fq(C)36 b(6)p Ft(=)29 b Fq(C)9 4686 y Fm(i)72 4671 y Ft(another)34 b(maximal)c(c)m(hain.)47 b(Since)34 b(t)m(w)m(o)h(p)s(oin)m(ts)e(in)g(the)h(same)g(maximal)c(c)m(hain)k (are)g(related,)g(at)f(most)-236 4810 y(one)g(of)f(the)h Fr(p)271 4825 y Fm(i)299 4810 y Ft('s)g(can)g(b)s(elong)e(to)i Fq(C)6 b Ft(.)-137 4950 y(If)35 b Fr(p)12 4965 y Fm(i)84 4950 y Fr(=)-60 b Fq(2)32 b(C)41 b Ft(for)35 b(all)e Fr(i)f Ft(=)g(1)p Fr(;)17 b(:)g(:)g(:)f(;)h(l)37 b Ft(then,)f(w)m(e)h (can)e(c)m(ho)s(ose)h Fr(p)2002 4965 y Fm(l)q Fp(+1)2150 4950 y Fq(2)d(C)6 b(n)17 b Ft(\()2411 4875 y Fh(S)2494 4979 y Fm(i)2539 4950 y Fq(C)2591 4965 y Fm(i)2620 4950 y Ft(\))35 b(and)g(it)f(will)f(not)i(b)s(e)g(related)g(to)-236 5089 y(an)m(y)e(of)f(the)h Fr(p)276 5104 y Fm(i)337 5089 y Ft(b)m(y)g(Remark)839 5089 y SDict begin H.S end 839 5089 a Ft(4.1)964 5025 y SDict begin H.R end 964 5025 a 964 5089 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (obs.4.1) cvn H.B /ANN pdfmark end 964 5089 a Ft(.)-137 5229 y(If)j Fr(p)13 5244 y Fm(i)74 5229 y Fq(2)d(C)42 b Ft(for)35 b(some)h(1)d Fq(\024)g Fr(i)g Fq(\024)h Fr(l)j Ft(then,)h(w)m(e)e(can)g (tak)m(e)h Fr(p)1948 5193 y Fk(0)1948 5254 y Fm(i)2009 5229 y Fq(2)c(C)2160 5244 y Fm(i)2189 5229 y Fq(nC)41 b Ft(and)36 b Fr(p)2574 5244 y Fm(l)q Fp(+1)2723 5229 y Fq(2)e(C)6 b(nC)2983 5244 y Fm(i)3047 5229 y Ft(not)35 b(related.)53 b(So,)36 b(the)-236 5368 y(p)s(oin)m(ts)c(in)g Fq(f)p Fr(p)270 5383 y Fp(1)309 5368 y Fr(;)17 b(:)g(:)g(:)f(;)h(p)577 5332 y Fk(0)577 5393 y Fm(i)605 5368 y Fr(;)g(:)g(:)g(:)e(p)828 5383 y Fm(l)854 5368 y Fr(;)i(p)947 5383 y Fm(l)q Fp(+1)1063 5368 y Fq(g)33 b Ft(will)d(b)s(e)j(pairwise)f(not)g(related)g(again)f (b)m(y)j(Remark)3104 5368 y SDict begin H.S end 3104 5368 a Ft(4.1)3229 5304 y SDict begin H.R end 3229 5304 a 3229 5368 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (obs.4.1) cvn H.B /ANN pdfmark end 3229 5368 a Ft(.)-137 5508 y(This)f(leads)g(us)h(to)e(a)h(con)m(tradiction)f(since)h(Lemma)1839 5508 y SDict begin H.S end 1839 5508 a Ft(3.3)1964 5444 y SDict begin H.R end 1964 5444 a 1964 5508 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lema.3.3) cvn H.B /ANN pdfmark end 1964 5508 a 33 w Ft(implies)d(the)k(existence)g(of)e Fr(\016)h(>)28 b Ft(0)k(and)h Fr(\027)3528 5472 y Fk(0)3580 5508 y Fq(2)c Ft(\(0)p Fr(;)17 b(\027)6 b Ft(\))-236 5647 y(suc)m(h)34 b(that)e(if)g Fr(p;)17 b(q)31 b Fq(2)d Fr(X)627 5662 y Fm(\016)697 5647 y Ft(then)1061 5852 y(dist)o(\()p Fr(S)1316 5867 y Fm(p)1378 5852 y Fq(\\)23 b Fr(@)5 b(B)1597 5867 y Fm(\027)1636 5848 y Ff(0)1663 5852 y Ft(\()p Fr(x)p Ft(\))p Fr(;)17 b(U)1904 5867 y Fm(q)1965 5852 y Fq(\\)22 b Fr(@)5 b(B)2183 5867 y Fm(\027)2222 5848 y Ff(0)2250 5852 y Ft(\()p Fr(x)p Ft(\)\))28 b Fr(>)f(\032)p eop end end %%Page: 11 11 TeXDict begin HPSdict begin 11 10 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a -245 -8 a SDict begin H.S end -245 -8 a -245 -8 a SDict begin H.R end -245 -8 a -245 -8 a SDict begin [/View [/XYZ H.V]/Dest (page.11) cvn /DEST pdfmark end -245 -8 a 475 66 a Fn(LOCAL)33 b(PR)n(ODUCT)g(STR)n(UCTURE)g(F)n(OR)f(EXP)-6 b(ANSIVE)32 b(HOMEOMORPHISMS)633 b(11)-137 266 y Ft(Giv)m(en)29 b Fr(p)191 281 y Fm(i)247 266 y Fq(2)f Fr(X)422 281 y Fm(\016)460 266 y Ft(,)i(Lemma)862 266 y SDict begin H.S end 862 266 a Ft(3.1)986 201 y SDict begin H.R end 986 201 a 986 266 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lema.3.1) cvn H.B /ANN pdfmark end 986 266 a 30 w Ft(ensures)h(the)e(existence)i (of)e Fr(q)2082 281 y Fm(i)2138 266 y Fq(2)36 b Ft(^)-57 b Fr(p)15 b Fq(\\)g Fr(@)5 b(B)2507 281 y Fm(\027)2546 262 y Ff(0)2574 266 y Ft(\()p Fr(x)p Ft(\))15 b Fq(\\)g Fr(U)2867 281 y Fm(p)2903 291 y Fi(i)2963 266 y Ft(so)30 b(that)f(dist)o(\()p Fr(q)3526 281 y Fm(i)3555 266 y Fr(;)17 b(q)3642 281 y Fm(j)3678 266 y Ft(\))28 b Fr(>)f(\032)-236 405 y Ft(if)35 b Fr(i)g Fq(6)p Ft(=)g Fr(j)6 b Ft(.)56 b(So,)38 b(there)f(exists)h(a)e(b)s(ound)h(on)g(the)g(n)m(um)m(b)s(er)g (of)f(pairwise)g(not)h(related)f(p)s(oin)m(ts)g(since)i Fr(@)5 b(B)3598 420 y Fm(\027)3637 401 y Ff(0)3664 405 y Ft(\()p Fr(x)p Ft(\))37 b(is)-236 545 y(compact.)-137 684 y(Once)45 b(w)m(e)f(kno)m(w)h(there)f(are)f(\014nitely)g(man)m(y)h (maximal)c(c)m(hains,)47 b(w)m(e)d(kno)m(w)h(that)e(the)h(ones)g(that)f (do)h(not)-236 824 y(accum)m(ulate)37 b(in)g Fr(x)h Ft(are)g(at)f(a)h (p)s(ositiv)m(e)f(distance)h(of)g Fr(x)p Ft(,)h(so)f(if)f(w)m(e)h(c)m (ho)s(ose)h Fr(\016)j Ft(to)37 b(b)s(e)h(smaller,)f(w)m(e)i(obtain)e (that)-236 963 y(ev)m(ery)d(maximal)c(c)m(hain)i(in)g Fr(B)865 978 y Fm(\016)903 963 y Ft(\()p Fr(x)p Ft(\))h(accum)m(ulates) f(in)g Fr(x)p Ft(.)-137 1103 y(Let)f Fr(C)36 b Ft(and)30 b Fr(C)406 1066 y Fk(0)460 1103 y Ft(b)s(e)g(t)m(w)m(o)g(maximal)d(c)m (hains)j(and)g Fr(q)i Fq(2)c Fr(C)c Fq(\\)17 b Fr(C)2070 1066 y Fk(0)2093 1103 y Ft(.)43 b(If)30 b(w)m(e)h(c)m(ho)s(ose)g Fr(\016)i Ft(smaller)28 b(in)i(suc)m(h)h(a)f(w)m(a)m(y)h(that)-229 1242 y(^)-56 b Fr(q)37 b Ft(b)s(e)d(disjoin)m(t)f(with)g Fr(B)624 1257 y Fm(\016)662 1242 y Ft(\()p Fr(x)p Ft(\),)h(w)m(e)h (reduce)g(the)f(n)m(um)m(b)s(er)g(of)f(maximal)e(c)m(hains)j(in)f Fr(B)2832 1257 y Fm(\016)2870 1242 y Ft(\()p Fr(x)p Ft(\).)47 b(So,)34 b(w)m(e)g(can)g(supp)s(ose)-236 1381 y(that)e(the)h(maximal)c (c)m(hains)k(are)g(pairwise)f(disjoin)m(t.)3820 1521 y Fj(\003)-137 1660 y Ft(W)-8 b(e)33 b(call)e([)p Fr(p)p Ft(])i(to)f(the)h(maximal)c(c)m(hain)k(of)f Fr(p)g Ft(in)g Fr(X)1676 1675 y Fm(\016)1747 1660 y Ft(giv)m(en)g(b)m(y)i(the)f (previous)g(Lemma.)42 b(No)m(w,)33 b(w)m(e)h(de\014ne)1395 1899 y Fr(S)1455 1915 y Fp([)p Fm(p)p Fp(])1562 1899 y Ft(=)p 1665 1783 220 4 v 1688 1805 a Fh([)1665 2020 y Fm(q)r Fk(2)p Fp([)p Fm(p)p Fp(])1845 1899 y Ft(^)-56 b Fr(q)31 b Fq(\032)p 2017 1813 249 4 v 28 w Fr(B)2091 1914 y Fm(\027)2135 1899 y Ft(\()p Fr(x)p Ft(\))-236 2206 y(where)j Fr(p)27 b Fq(2)h Fr(X)297 2221 y Fm(\016)335 2206 y Ft(.)-137 2345 y(W)-8 b(e)28 b(remark)f(that)f(w)m(e)j(can)e(c)m (ho)s(ose)h Fr(\027)33 b Ft(suc)m(h)c(that)e(for)f(ev)m(ery)j Fr(p)f Fq(2)g Fr(P)14 b(er)2409 2360 y Fm(H)2486 2345 y Fq(\\)d Fr(B)2637 2360 y Fm(\027)2681 2345 y Ft(\()p Fr(x)p Ft(\))27 b(w)m(e)h(ha)m(v)m(e)h(that)d Fr(S)3462 2360 y Fm(p)3530 2345 y Fq(2)i Fr(W)3730 2309 y Fm(s)3716 2370 y(")3766 2345 y Ft(\()p Fr(x)p Ft(\))-236 2485 y(where)34 b(0)27 b Fr(<)h Ft(2)p Fr(")f(<)g(\013)33 b Ft(and)g Fr(\013)28 b(>)g Ft(0)k(is)g(the)h(constan)m(t)h(of)e(expansivit)m(y)-8 b(.)-236 2715 y SDict begin H.S end -236 2715 a -236 2715 a SDict begin 16.79996 H.A end -236 2715 a -236 2715 a SDict begin [/View [/XYZ H.V]/Dest (lema.4.2) cvn /DEST pdfmark end -236 2715 a Fy(Lemma)37 b(4.2.)42 b Fl(F)-7 b(or)34 b(every)g(maximal)g(chain)g Ft([)p Fr(p)p Ft(])p Fl(,)h Fr(@)1715 2604 y Fh(\020)1775 2640 y(S)1858 2744 y Fm(q)r Fk(2)p Fp([)p Fm(p)p Fp(])2042 2715 y Ft(^)-57 b Fr(q)2081 2604 y Fh(\021)2163 2715 y Fq(\\)22 b Fr(B)2325 2730 y Fm(\027)2369 2715 y Ft(\()p Fr(x)p Ft(\))28 b Fq(\032)g Fr(W)2739 2679 y Fm(s)2725 2740 y(")2775 2715 y Ft(\()p Fr(x)p Ft(\))35 b Fl(veri\014es.)-236 2981 y Fs(Pr)n(oof.)82 b Ft(Because)28 b(of)d(Lemma)1001 2981 y SDict begin H.S end 1001 2981 a Ft(4.1)1126 2917 y SDict begin H.R end 1126 2917 a 1126 2981 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lema.4.1) cvn H.B /ANN pdfmark end 1126 2981 a 25 w Ft(w)m(e)i(kno)m(w)g(that)f([)p Fr(p)p Ft(])f(accum)m(ulates)h (in)f Fr(x)p Ft(.)42 b(Let)25 b Fr(q)2854 2996 y Fm(n)2929 2981 y Fq(2)j Ft([)p Fr(p)p Ft(])e(suc)m(h)h(that)f Fr(q)3613 2996 y Fm(n)3688 2981 y Fq(!)h Fr(x)p Ft(.)-236 3129 y(T)-8 b(ak)m(e)33 b(a)e(p)s(oin)m(t)g Fr(y)g Fq(2)d Fr(@)579 3018 y Fh(\020)639 3054 y(S)722 3158 y Fm(q)r Fk(2)p Fp([)p Fm(p)p Fp(])905 3129 y Ft(^)-56 b Fr(q)945 3018 y Fh(\021)1005 3129 y Ft(.)43 b(Then,)33 b(a)e(sequence)k Fr(z)1885 3144 y Fm(n)1960 3129 y Fq(2)g Ft(^)-56 b Fr(q)2097 3144 y Fm(n)2176 3129 y Ft(exists)32 b(suc)m(h)h(that)f Fr(z)2919 3144 y Fm(n)2994 3129 y Fq(!)27 b Fr(y)t Ft(.)43 b(Without)31 b(loss)g(of)-236 3271 y(generalit)m(y)h(w)m(e)h(can)g (supp)s(ose)h Fr(z)948 3286 y Fm(n)1023 3271 y Fq(2)28 b Fr(S)1177 3286 y Fm(q)1209 3294 y Fi(n)1255 3271 y Ft(.)-137 3410 y(Remark)237 3410 y SDict begin H.S end 237 3410 a Ft(1.1)362 3346 y SDict begin H.R end 362 3346 a 362 3410 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (obs.1.1) cvn H.B /ANN pdfmark end 362 3410 a 40 w Ft(ensures)41 b(the)f(existence)h(of)e Fr(")g(>)g Ft(0)g(suc)m(h)i(that)f Fr(S)2260 3425 y Fm(q)2292 3433 y Fi(n)2377 3410 y Fq(\032)g Fr(W)2600 3374 y Fm(s)2586 3435 y(")2636 3410 y Ft(\()p Fr(z)2723 3374 y Fk(0)2719 3435 y Fm(n)2767 3410 y Ft(\).)64 b(So)39 b(w)m(e)i(ha)m(v)m(e)f(that)g(for)f(all)-236 3550 y Fr(m)28 b Fq(\025)g Ft(0)749 3861 y(dist)o(\()p Fr(f)1003 3820 y Fm(m)1070 3861 y Ft(\()p Fr(y)t Ft(\))p Fr(;)17 b(f)1301 3820 y Fm(m)1366 3861 y Ft(\()p Fr(x)p Ft(\)\))27 b(=)52 b(lim)1666 3921 y Fm(n)p Fk(!1)1867 3861 y Ft(dist)o(\()p Fr(f)2121 3820 y Fm(m)2187 3861 y Ft(\()p Fr(z)2274 3820 y Fk(0)2270 3886 y Fm(n)2318 3861 y Ft(\))p Fr(;)17 b(f)2459 3820 y Fm(m)2525 3861 y Ft(\()p Fr(p)2612 3876 y Fm(n)2658 3861 y Ft(\)\))28 b Fq(\024)g Fr(")-236 4110 y Ft(so)33 b Fr(y)d Fq(2)e Fr(W)162 4074 y Fm(s)148 4135 y(")199 4110 y Ft(\()p Fr(x)p Ft(\).)44 b(Then,)33 b Fr(@)756 4000 y Fh(\020)815 4036 y(S)898 4140 y Fm(q)r Fk(2)p Fp([)p Fm(p)p Fp(])1082 4110 y Ft(^)-56 b Fr(q)1122 4000 y Fh(\021)1209 4110 y Fq(\032)28 b Fr(W)1420 4074 y Fm(s)1406 4135 y(")1456 4110 y Ft(\()p Fr(x)p Ft(\).)3820 4251 y Fj(\003)-236 4478 y SDict begin H.S end -236 4478 a -236 4478 a SDict begin 16.79996 H.A end -236 4478 a -236 4478 a SDict begin [/View [/XYZ H.V]/Dest (lema.4.3) cvn /DEST pdfmark end -236 4478 a Fy(Lemma)d(4.3.)33 b Fl(Supp)-5 b(ose)p 718 4398 233 4 v 25 w Fr(P)14 b(er)884 4493 y Fm(H)978 4478 y Ft(=)28 b Fr(M)36 b Fl(and)24 b(let)i Fr(x)i Fq(2)g Fr(M)36 b Fl(b)-5 b(e)25 b(a)f(singularity.)42 b(Then,)26 b(for)f(al)5 b(l)25 b Fr(p)j Fq(2)g Fr(B)3326 4493 y Fm(\016)3364 4478 y Ft(\()p Fr(x)p Ft(\))q Fq(\\)q Fr(P)14 b(er)3732 4437 y Fm(n)p Fk(\000)p Fp(1)3729 4505 y Fm(H)3868 4478 y Fl(,)-236 4618 y(ther)-5 b(e)35 b(exists)f(a)h(neighb)-5 b(orho)g(o)g(d)33 b Fr(V)56 b Fl(of)35 b Fr(S)1225 4633 y Fm(p)1299 4618 y Fl(such)g(that)g Fr(P)14 b(er)1886 4633 y Fm(H)1975 4618 y Fq(\\)23 b Fr(V)44 b Fq(\\)22 b Fr(B)2327 4633 y Fm(\016)2365 4618 y Ft(\()p Fr(x)p Ft(\))28 b Fq(\032)g Ft([)p Fr(p)p Ft(])p Fl(.)-236 4876 y Fs(Pr)n(oof.)45 b Ft(By)33 b(con)m(tradiction,)f(let)h(us)g(supp)s (ose)i(that)d(there)i(exists)g Fr(y)d Fq(2)e Fr(S)2526 4891 y Fm(p)2588 4876 y Fq(\\)23 b Fr(B)2751 4891 y Fm(\016)2789 4876 y Ft(\()p Fr(x)p Ft(\))33 b(satisfying)f(that)h Fr(q)3643 4891 y Fm(n)3719 4876 y Fq(!)27 b Fr(y)-236 5015 y Ft(with)46 b Fr(q)43 5030 y Fm(n)141 5015 y Fq(2)52 b Ft([)p Fr(q)t Ft(])f Fq(6)p Ft(=)g([)p Fr(p)p Ft(])c(\(remem)m(b)s (er)f(that)g(b)s(ecause)i(of)e(Theorem)2346 5015 y SDict begin H.S end 2346 5015 a Ft(1.1)2470 4951 y SDict begin H.R end 2470 4951 a 2470 5015 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (teo.1.1) cvn H.B /ANN pdfmark end 2470 5015 a Ft(,)k(near)d Fr(S)2838 5030 y Fm(p)2924 5015 y Ft(w)m(e)g(ha)m(v)m(e)h(lo)s(cal)c(pro)s(duct)-236 5155 y(structure)32 b(so)e(ev)m(ery)j(p)s(erio)s(dic)c(p)s(oin)m(t)h (near)h Fr(y)i Ft(m)m(ust)e(ha)m(v)m(e)h(the)f(same)f(index)h(as)g Fr(p)p Ft(\).)43 b(Then)32 b Fr(y)e Fq(2)f Fr(@)5 b(S)3423 5170 y Fp([)p Fm(q)r Fp(])3531 5155 y Ft(\(b)s(ecause)-236 5294 y(it)28 b(b)s(elongs)g(b)s(oth)h(to)f Fr(int)p Ft(\()p Fr(S)770 5310 y Fp([)p Fm(p)p Fp(])850 5294 y Ft(\))h(and)g Fr(S)1163 5310 y Fp([)p Fm(q)r Fp(])1240 5294 y Ft(,)h(and)f(the)g(in)m (teriors)f(of)h Fr(S)2194 5310 y Fp([)p Fm(p)p Fp(])2301 5294 y Ft(and)h Fr(S)2548 5310 y Fp([)p Fm(q)r Fp(])2654 5294 y Ft(ha)m(v)m(e)g(empt)m(y)f(in)m(tersection\).)43 b(So,)-236 5434 y(b)m(y)35 b(Lemma)251 5434 y SDict begin H.S end 251 5434 a Ft(4.2)376 5369 y SDict begin H.R end 376 5369 a 376 5434 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lema.4.2) cvn H.B /ANN pdfmark end 376 5434 a Ft(,)g Fr(y)f Fq(2)e Fr(W)724 5397 y Fm(s)710 5458 y(")760 5434 y Ft(\()p Fr(x)p Ft(\).)50 b(Therefore)36 b Fr(x)31 b Fq(2)h Fr(W)1702 5397 y Fm(s)1738 5434 y Ft(\()p Fr(p)p Ft(\))i(b)s(ecause)i Fr(y)f Fq(2)c Fr(S)2500 5449 y Fm(p)2571 5434 y Fq(\032)g Fr(W)2785 5397 y Fm(s)2771 5458 y(")2822 5434 y Ft(\()p Fr(p)p Ft(\).)49 b(But,)35 b(since)g Fr(p)c Fq(2)h Fr(P)14 b(er)3831 5449 y Fm(H)-236 5573 y Ft(w)m(e)35 b(con)m(tradict)f(the)g(fact)g(that)g Fr(x)g Ft(is)f(singular,)g(since) i(Theorem)2190 5573 y SDict begin H.S end 2190 5573 a Ft(1.1)2315 5509 y SDict begin H.R end 2315 5509 a 2315 5573 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (teo.1.1) cvn H.B /ANN pdfmark end 2315 5573 a 33 w Ft(giv)m(es)g(us)f(lo)s(cal)e (pro)s(duct)i(structure)i(in)d(a)-236 5712 y(neigh)m(b)s(orho)s(o)s(d)e (of)h Fr(x)h Ft(b)m(y)h(iteration)c(of)i(the)h(lo)s(cal)e(pro)s(duct)i (structure)g(in)f Fr(p)p Ft(.)3820 5852 y Fj(\003)p eop end end %%Page: 12 12 TeXDict begin HPSdict begin 12 11 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont 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Ft(\()p Fr(x)p Ft(\))p Fl(.)-236 640 y Fs(Pr)n(oof.)-137 779 y Ft(The)j(inclusion)466 704 y Fh(S)549 808 y Fm(q)r Fk(2)p Fp([)p Fm(p)p Fp(])732 779 y Ft(^)-56 b Fr(q)20 b Fq(\\)d Fr(B)945 794 y Fm(\016)983 779 y Ft(\()p Fr(x)p Ft(\))28 b Fq(\032)h Ft(in)m(t)o(\()p Fr(S)1462 795 y Fp([)p Fm(p)p Fp(])1541 779 y Ft(\))17 b Fq(\\)g Fr(B)1753 794 y Fm(\016)1790 779 y Ft(\()p Fr(x)p Ft(\))30 b(is)g(immediate)d(b)s (ecause)k(if)e Fr(q)i Fq(\025)e Fr(r)j Ft(then)j(^)-53 b Fr(r)30 b Fq(\032)e Fr(int)p Ft(\()7 b(^)-56 b Fr(q)5 b Ft(\).)-137 919 y(T)-8 b(o)38 b(obtain)e(the)h(other)g(inclusion)f(w) m(e)i(pro)s(ceed)g(b)m(y)g(con)m(tradiction)e(supp)s(osing)h(there)h (exists)g(a)e(p)s(oin)m(t)h Fr(y)h Fq(2)-236 1058 y Fr(B)-162 1073 y Fm(\016)-124 1058 y Ft(\()p Fr(x)p Ft(\))33 b(in)e(the)i(in)m (terior)f(of)g Fr(S)837 1074 y Fp([)p Fm(p)p Fp(])948 1058 y Ft(but)h(suc)m(h)h(that)e Fr(y)43 b(=)-61 b Fq(2)35 b Ft(^)-56 b Fr(q)36 b Ft(for)d(all)d Fr(q)i Fq(2)c Ft([)p Fr(p)p Ft(].)-137 1198 y(Then,)i(there)f(exists)g Fr(y)699 1213 y Fm(n)773 1198 y Fq(2)f Fr(S)927 1213 y Fm(q)959 1221 y Fi(n)1033 1198 y Ft(suc)m(h)i(that)d Fr(y)1503 1213 y Fm(n)1577 1198 y Fq(!)h Fr(y)j Ft(\(this)c(implies)f(in)h (particular)f(that)i Fr(y)j Fq(2)d Fr(W)3374 1161 y Fm(s)3360 1222 y(")3410 1198 y Ft(\()p Fr(x)p Ft(\))g(b)s(ecause)-236 1337 y(of)k(Lemma)223 1337 y SDict begin H.S end 223 1337 a Ft(3.2)348 1273 y SDict begin H.R end 348 1273 a 348 1337 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lema.3.2) cvn H.B /ANN pdfmark end 348 1337 a Ft(\))g(where)i Fr(q)743 1352 y Fm(n)818 1337 y Fq(2)28 b Ft([)p Fr(p)p Ft(])33 b(satis\014es)g Fr(q)1450 1352 y Fm(n)1525 1337 y Fq(!)27 b Fr(x)p Ft(.)-137 1477 y(Using)47 b(Lemma)514 1477 y SDict begin H.S end 514 1477 a Ft(4.3)639 1412 y SDict begin H.R end 639 1412 a 639 1477 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lema.4.3) cvn H.B /ANN pdfmark end 639 1477 a 46 w Ft(and)g(the)g(fact)g(that)p 1503 1397 233 4 v 46 w Fr(P)14 b(er)1669 1492 y Fm(H)1787 1477 y Ft(=)52 b Fr(M)57 b Ft(w)m(e)48 b(kno)m(w)g(that)e(there)i(exist)f(p)s(oin)m (ts)f Fr(r)3578 1492 y Fm(n)3676 1477 y Fq(2)52 b Ft([)p Fr(p)p Ft(])-236 1616 y(arbitrarily)41 b(close)i(to)g Fr(y)666 1631 y Fm(n)713 1616 y Ft(.)76 b(W)-8 b(e)44 b(can)g(supp)s(ose)h Fr(r)1607 1631 y Fm(n)1701 1616 y Fq(!)h Fr(y)g Ft(and)e(that)f(this)h(p)s(oin)m(ts)f(are)h(not)f(b)s (ounded)h(in)f(the)-236 1756 y(ordering)h(in)f([)p Fr(p)p Ft(].)80 b(On)45 b(the)g(other)f(hand,)k(w)m(e)e(consider)f Fr(U)2017 1771 y Fm(r)2049 1779 y Fi(n)2144 1756 y Ft(=)i Fr(cc)2351 1771 y Fm(r)2383 1779 y Fi(n)2430 1756 y Ft(\()p Fr(B)2542 1771 y Fm(\027)2585 1756 y Ft(\()p Fr(x)p Ft(\))31 b Fq(\\)f Fr(W)2949 1719 y Fm(u)2994 1756 y Ft(\()p Fr(r)3076 1771 y Fm(n)3123 1756 y Ft(\)\))48 b Fq(\032)h Fr(W)3479 1719 y Fm(u)3465 1780 y(")3523 1756 y Ft(\()p Fr(r)3605 1771 y Fm(n)3652 1756 y Ft(\))c(\(see)-236 1895 y(Remark)130 1895 y SDict begin H.S end 130 1895 a Ft(1.1)254 1831 y SDict begin H.R end 254 1831 a 254 1895 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (obs.1.1) cvn H.B /ANN pdfmark end 254 1895 a Ft(\))32 b(whic)m(h)g(is)f(separated)h(b)m(y)g Fr(S)1331 1910 y Fm(r)1363 1918 y Fi(n)1441 1895 y Ft(in)f(t)m(w)m(o)h (di\013eren)m(t)g(connected)h(comp)s(onen)m(ts)f(\(see)g(corollary)3708 1895 y SDict begin H.S end 3708 1895 a Ft(3.1)3833 1831 y SDict begin H.R end 3833 1831 a 3833 1895 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cor.3.1) cvn H.B /ANN pdfmark end 3833 1895 a Ft(\).)-137 2034 y(Pic)m(k)37 b Fr(\015)j(>)34 b Ft(0)i(and)h(c)m(ho)s(ose)g Fr(z)923 2049 y Fm(n)1005 2034 y Fq(2)e Fr(@)5 b(B)1236 2049 y Fm(\015)1281 2034 y Ft(\()p Fr(y)t Ft(\))24 b Fq(\\)h Fr(U)1590 2049 y Fm(r)1622 2057 y Fi(n)1706 2034 y Ft(suc)m(h)38 b(that)e Fr(z)2190 2049 y Fm(n)2283 2034 y Fr(=)-60 b Fq(2)39 b Ft(^)-54 b Fr(r)2416 2049 y Fm(n)2463 2034 y Ft(.)56 b(W)-8 b(e)36 b(can)h(supp)s(ose)h(that)e Fr(z)3531 2049 y Fm(n)3613 2034 y Fq(!)e Fr(z)39 b Fq(2)-236 2174 y Fr(@)5 b(B)-106 2189 y Fm(\015)-61 2174 y Ft(\()p Fr(y)t Ft(\))33 b(and)g(using)g(the)h(semicon)m(tin)m(uous)f(v)-5 b(ariation)31 b(of)i(lo)s(cal)e(stable)i(and)h(unstable)f(sets)i (\(Lemma)3591 2174 y SDict begin H.S end 3591 2174 a Ft(3.2)3715 2110 y SDict begin H.R end 3715 2110 a 3715 2174 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lema.3.2) cvn H.B /ANN pdfmark end 3715 2174 a Ft(\))f(w)m(e)-236 2313 y(obtain)d(that)i Fr(z)f Fq(2)c Fr(W)556 2277 y Fm(u)542 2338 y(")601 2313 y Ft(\()p Fr(y)t Ft(\).)-137 2453 y(W)-8 b(e)42 b(shall)e(pro)m(v)m(e)j(that)e Fr(z)60 b(=)-61 b Fq(2)43 b Fr(S)1031 2468 y Fp([)p Fm(p)p Fp(])1152 2453 y Ft(and)e(since)h Fr(\015)k Ft(w)m(as)d(arbitrary)d(this)i(will)d (imply)g(that)j Fr(y)k Fq(2)d Fr(@)5 b(S)3530 2468 y Fp([)p Fm(p)p Fp(])3651 2453 y Ft(whic)m(h)-236 2592 y(con)m(tradicts)33 b(the)g(fact)f(that)g Fr(y)f Fq(2)d Fr(int)p Ft(\()p Fr(S)1234 2608 y Fp([)p Fm(p)p Fp(])1314 2592 y Ft(\).)-137 2732 y(W)-8 b(e)39 b(kno)m(w)h(that)f Fr(z)54 b(=)-61 b Fq(2)46 b Ft(^)-56 b Fr(q)42 b Ft(for)c(all)f Fr(q)k Fq(2)e Ft([)p Fr(p)p Ft(],)h(so,)h(if)c Fr(z)43 b Fq(2)38 b Fr(S)1951 2747 y Fp([)p Fm(p)p Fp(])2069 2732 y Ft(it)f(should)i(b)s(e)f(accum)m(ulated)h(b)m(y)g(p)s(oin)m(ts)g (in)e Fr(S)3819 2747 y Fm(q)3851 2755 y Fi(n)-236 2871 y Ft(and)i(therefore)i(v)m(erify)f Fr(z)k Fq(2)c Fr(W)956 2835 y Fm(s)942 2896 y(")992 2871 y Ft(\()p Fr(x)p Ft(\).)65 b(Then,)43 b Fr(z)h Fq(6)p Ft(=)39 b Fr(y)t Ft(,)i Fr(z)j Fq(2)c Fr(W)2131 2835 y Fm(u)2117 2896 y(")2176 2871 y Ft(\()p Fr(y)t Ft(\))e(and)i Fr(y)t(;)17 b(z)43 b Fq(2)d Fr(W)2935 2835 y Fm(s)2921 2896 y(")2971 2871 y Ft(\()p Fr(x)p Ft(\))g(whic)m(h)g(con)m(tradicts)-236 3011 y(expansivit)m(y)33 b(\(remem)m(b)s(er)f(w)m(e)i(c)m(hose)g Fr(")e Ft(so)h(that)f(2)p Fr(")27 b(<)h(\013)q Ft(\).)3820 3150 y Fj(\003)-236 3350 y SDict begin H.S end -236 3350 a -236 3350 a SDict begin 16.79996 H.A end -236 3350 a -236 3350 a SDict begin [/View [/XYZ H.V]/Dest (obs.4.2) cvn /DEST pdfmark end -236 3350 a Fl(R)-5 b(emark)47 b Ft(4.2)p Fl(.)c Ft(Clearly)35 b Fr(x)e Fq(2)g Fr(S)925 3365 y Fp([)p Fm(p)p Fp(])1039 3350 y Ft(and)i Fr(x)45 b(=)-61 b Fq(2)41 b Ft(^)-57 b Fr(q)40 b Ft(for)35 b(all)e Fr(q)j Fq(2)d Ft([)p Fr(p)p Ft(].)53 b(Since)35 b Fr(S)2469 3365 y Fp([)p Fm(p)p Fp(])2583 3350 y Ft(is)g(a)h(closed)f(set)h(with)g(non)f(empt)m(y)-236 3489 y(in)m(terior)26 b(and)i Fr(x)g Fq(2)h Fr(@)5 b(S)582 3505 y Fp([)p Fm(p)p Fp(])689 3489 y Ft(w)m(e)29 b(ha)m(v)m(e)h(that)d (its)h(complemen)m(t)f(in)g Fr(B)2110 3504 y Fm(\027)2153 3489 y Ft(\()p Fr(x)p Ft(\))h(whic)m(h)h(is)e(op)s(en)i(is)e(also)g (non)h(empt)m(y)-8 b(.)42 b(This)-236 3629 y(implies)30 b(that)i Fr(@)5 b(S)422 3644 y Fp([)p Fm(p)p Fp(])535 3629 y Ft(separates)33 b Fr(B)1032 3644 y Fm(\027)1076 3629 y Ft(\()p Fr(x)p Ft(\).)-137 3828 y(The)f(next)f(lemma)d(sho)m(ws) k(ho)m(w)f(the)f(stable)g(sets)i(of)e(p)s(erio)s(dic)e(p)s(oin)m(ts)i (con)m(v)m(erge)i(uniformly)c(to)m(w)m(ards)k Fr(@)5 b(S)3791 3844 y Fp([)p Fm(p)p Fp(])3871 3828 y Ft(.)-236 4028 y SDict begin H.S end -236 4028 a -236 4028 a SDict begin 16.79996 H.A end -236 4028 a -236 4028 a SDict begin [/View [/XYZ H.V]/Dest (lema.4.5) cvn /DEST pdfmark end -236 4028 a Fy(Lemma)42 b(4.5.)i Fl(Supp)-5 b(ose)p 759 3948 233 4 v 38 w Fr(P)14 b(er)925 4043 y Fm(H)1027 4028 y Ft(=)35 b Fr(M)50 b Fl(and)39 b(let)g Fr(z)h Fq(2)35 b Fr(@)5 b(S)1919 4043 y Fp([)p Fm(p)p Fp(])2024 4028 y Fq(\\)26 b Fr(B)2190 4043 y Fm(\016)2228 4028 y Ft(\()p Fr(x)p Ft(\))39 b Fl(and)f Fr(\032)e(>)f Ft(0)p Fl(.)57 b(Then,)39 b(ther)-5 b(e)39 b(exists)f Fr(V)61 b Fl(a)-236 4167 y(neighb)-5 b(orho)g(o)g(d)33 b(of)h Fr(z)40 b Fl(such)35 b(that)g(if)g Fr(q)c Fq(2)d Ft([)p Fr(p)p Ft(])23 b Fq(\\)f Fr(V)56 b Fl(then)35 b Fr(S)1830 4183 y Fp([)p Fm(p)p Fp(])1931 4167 y Fq(\\)p 2020 4081 244 4 v 23 w Fr(B)2094 4182 y Fm(\016)2132 4167 y Ft(\()p Fr(x)p Ft(\))28 b Fq(\032)35 b Ft(^)-56 b Fr(q)26 b Fq([)c Fr(B)2627 4182 y Fm(\032)2668 4167 y Ft(\()p Fr(@)5 b(S)2822 4183 y Fp([)p Fm(p)p Fp(])2902 4167 y Ft(\))p Fl(.)-236 4397 y Fs(Pr)n(oof.)148 b Ft(Giv)m(en)43 b Fr(\032)i(>)g Ft(0,)g(the)e(set)g Fr(K)52 b Ft(=)1481 4316 y Fh(\000)1527 4397 y Fr(S)1587 4413 y Fp([)p Fm(p)p Fp(])1688 4397 y Fq(n)22 b Fr(B)1834 4412 y Fm(\032)1874 4397 y Ft(\()p Fr(@)5 b(S)2028 4413 y Fp([)p Fm(p)p Fp(])2108 4397 y Ft(\))2146 4316 y Fh(\001)2221 4397 y Fq(\\)p 2316 4310 V 29 w Fr(B)2390 4412 y Fm(\016)2428 4397 y Ft(\()p Fr(x)p Ft(\))43 b(is)f(a)h(compact)f(set)i(con)m(tained) e(in)-236 4537 y(in)m(t)o(\()p Fr(S)-22 4552 y Fp([)p Fm(p)p Fp(])83 4537 y Fq(\\)26 b Fr(B)249 4552 y Fm(\016)287 4537 y Ft(\()p Fr(x)p Ft(\)\))39 b(so,)h(using)e(Lemma)1264 4537 y SDict begin H.S end 1264 4537 a Ft(4.4)1388 4472 y SDict begin H.R end 1388 4472 a 1388 4537 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lema.4.4) cvn H.B /ANN pdfmark end 1388 4537 a Ft(,)i Fq(f)p Ft(in)m(t)o(\()7 b(^)-56 b Fr(q)t Ft(\))p Fq(g)1794 4552 y Fm(q)r Fk(2)p Fp([)p Fm(p)p Fp(])1992 4537 y Ft(is)37 b(an)i(op)s(en)f(co)m(v)m(er)i(of)d Fr(K)46 b Ft(so)38 b Fr(r)i Fq(2)e Ft([)p Fr(p)p Ft(])g(exists)h(suc)m (h)-236 4676 y(that)34 b Fr(K)39 b Fq(\032)d Ft(^)-53 b Fr(r)r Ft(.)50 b(Let)35 b Fr(V)57 b Ft(b)s(e)35 b(a)f(neigh)m(b)s (orho)s(o)s(d)g(of)g Fr(z)39 b Ft(disjoin)m(t)34 b(from)k(^)-54 b Fr(r)s Ft(.)50 b(Then,)37 b(for)d(ev)m(ery)i Fr(q)g Fq(2)31 b Ft([)p Fr(p)p Ft(])24 b Fq(\\)g Fr(V)56 b Ft(w)m(e)36 b(ha)m(v)m(e)-236 4815 y(that)c Fr(q)g Fq(\025)c Fr(r)s Ft(.)43 b(Then,)34 b Fr(K)h Ft(=)775 4735 y Fh(\000)821 4815 y Fr(S)881 4831 y Fp([)p Fm(p)p Fp(])982 4815 y Fq(n)22 b Fr(B)1128 4830 y Fm(\032)1168 4815 y Ft(\()p Fr(@)5 b(S)1322 4831 y Fp([)p Fm(p)p Fp(])1402 4815 y Ft(\))1440 4735 y Fh(\001)1508 4815 y Fq(\\)p 1596 4729 244 4 v 22 w Fr(B)1670 4830 y Fm(\016)1708 4815 y Ft(\()p Fr(x)p Ft(\))28 b Fq(\032)35 b Ft(^)-56 b Fr(q)37 b Ft(and)32 b(therefore)h Fr(S)2710 4831 y Fp([)p Fm(p)p Fp(])2811 4815 y Fq(\\)p 2900 4729 V 23 w Fr(B)2974 4830 y Fm(\016)3012 4815 y Ft(\()p Fr(x)p Ft(\))28 b Fq(\032)35 b Ft(^)-56 b Fr(q)26 b Fq([)d Fr(B)3508 4830 y Fm(\032)3548 4815 y Ft(\()p Fr(@)5 b(S)3702 4831 y Fp([)p Fm(p)p Fp(])3782 4815 y Ft(\).)3820 4955 y Fj(\003)-137 5094 y Ft(The)39 b(follo)m(wing)c(Lemma)h(represen)m(ts)41 b(the)d(k)m(ey)h(step)g(for)e (pro)m(ving)g(the)i(uniformit)m(y)c(of)j(the)g(lo)s(cal)d(pro)s(duct) -236 5234 y(structure)e(b)s(ecause)h(it)d(allo)m(ws)h(us)h(to)f(ensure) h(that)f(the)h(stable)f(and)h(unstable)f(sets)i(in)m(tersect)f(in)f(a)g (neigh)m(b)s(or-)-236 5373 y(ho)s(o)s(d)j(of)g(a)h(singularit)m(y)-8 b(.)51 b(This)36 b(giv)m(es)g(uniformit)m(y)e(and)i(is)f(also)g(imp)s (ortan)m(t)e(to)j(giv)m(e)g(structure)h(to)e Fr(@)5 b(S)3625 5389 y Fp([)p Fm(p)p Fp(])3740 5373 y Ft(and)-236 5513 y(discard)32 b(singularities.)-236 5712 y SDict begin H.S end -236 5712 a -236 5712 a SDict begin 16.79996 H.A end -236 5712 a -236 5712 a SDict begin [/View [/XYZ H.V]/Dest (lema.4.6) cvn /DEST pdfmark end -236 5712 a Fy(Lemma)46 b(4.6.)g Fl(Supp)-5 b(ose)p 768 5632 233 4 v 41 w Fr(P)14 b(er)934 5727 y Fm(H)1042 5712 y Ft(=)40 b Fr(M)10 b Fl(.)67 b(F)-7 b(or)41 b(al)5 b(l)42 b Fr(z)k Fq(2)41 b Fr(@)5 b(S)2009 5728 y Fp([)p Fm(p)p Fp(])2116 5712 y Fq(\\)28 b Fr(B)2284 5727 y Fm(\016)2322 5712 y Ft(\()p Fr(x)p Ft(\))42 b Fl(and)g(for)g(al)5 b(l)41 b Fr(")g(>)g Ft(0)g Fl(ther)-5 b(e)42 b(exists)g Fr(V)-236 5852 y Fl(neighb)-5 b(orho)g(o)g(d)33 b(of)h Fr(z)40 b Fl(such)35 b(that)g(if)g Fr(q)t(;)17 b(r)30 b Fq(2)e Fr(V)43 b Fq(\\)23 b Ft([)p Fr(p)p Ft(])35 b Fl(then)g Fr(U)1927 5867 y Fm(q)2000 5852 y Fl(interse)-5 b(cts)34 b Fr(S)2494 5867 y Fm(r)2567 5852 y Fl(and)g Fr(@)5 b(S)2872 5867 y Fp([)p Fm(p)p Fp(])2987 5852 y Fl(in)35 b Fr(B)3181 5867 y Fm(")3218 5852 y Ft(\()p Fr(z)t Ft(\))22 b Fq(\\)h Fr(S)3514 5867 y Fp([)p Fm(p)p Fp(])3593 5852 y Fl(.)p eop end end %%Page: 13 13 TeXDict begin HPSdict begin 13 12 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a -245 -8 a SDict begin H.S end -245 -8 a -245 -8 a SDict begin H.R end -245 -8 a -245 -8 a SDict begin [/View [/XYZ H.V]/Dest (page.13) cvn /DEST pdfmark end -245 -8 a 475 66 a Fn(LOCAL)33 b(PR)n(ODUCT)g(STR)n(UCTURE)g(F)n(OR)f(EXP)-6 b(ANSIVE)32 b(HOMEOMORPHISMS)633 b(13)-236 266 y Fs(Pr)n(oof.)124 b Ft(Let)39 b Fr(V)61 b Ft(b)s(e)39 b(a)f(neigh)m(b)s(orho)s(o)s(d)g (of)g Fr(z)44 b Ft(suc)m(h)c(that)f Fr(z)k Fq(2)38 b Fr(V)60 b Fq(\032)39 b Fr(B)2524 281 y Fm(\016)2562 266 y Ft(\()p 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33 w Ft(w)m(e)g(can)g(supp)s(ose)h(that)f(for)f(ev)m(ery)i Fr(q)d Fq(2)e Ft([)p Fr(p)p Ft(])22 b Fq(\\)g Fr(V)g Ft(,)-236 1256 y(\(2\))1211 1256 y SDict begin H.S end 1211 1256 a 1211 1256 a SDict begin 16.79996 H.A end 1211 1256 a 1211 1256 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.2) cvn /DEST pdfmark end 1211 1256 a Fr(S)1271 1271 y Fp([)p Fm(p)p Fp(])1372 1256 y Fq(\\)g Fr(B)1534 1271 y Fm(\016)1572 1256 y Ft(\()p Fr(x)p Ft(\))28 b Fq(\032)36 b Ft(^)-57 b Fr(q)26 b Fq([)d Fr(B)2068 1271 y Fm(\032=)p Fp(2)2179 1256 y Ft(\()p Fr(@)5 b(S)2333 1271 y Fp([)p Fm(p)p Fp(])2413 1256 y Ft(\))-137 1446 y(Then,)40 b(since)d Fr(y)446 1399 y Fm(q)442 1471 y Fp(2)530 1446 y Fr(=)-61 b Fq(2)42 b Ft(^)-56 b Fr(q)41 b Ft(and)c Fr(y)949 1399 y Fm(q)945 1471 y Fp(2)1033 1446 y Fr(=)-61 b Fq(2)35 b Fr(B)1196 1461 y Fm(\032)1237 1446 y Ft(\()p Fr(@)5 b(S)1391 1462 y Fp([)p Fm(p)p Fp(])1471 1446 y Ft(\),)37 b(w)m(e)h(ha)m(v)m(e)g(that)f Fr(y)2218 1399 y Fm(q)2214 1471 y Fp(2)2302 1446 y Fr(=)-61 b Fq(2)36 b Fr(S)2452 1462 y Fp([)p Fm(p)p Fp(])2530 1446 y Ft(.)57 b(Remark)2985 1446 y SDict begin H.S end 2985 1446 a Ft(4.2)3110 1382 y SDict begin H.R end 3110 1382 a 3110 1446 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (obs.4.2) cvn H.B /ANN pdfmark end 3110 1446 a 37 w Ft(together)37 b(with)f(the)-236 1586 y(fact)c(that)h Fr(U)234 1601 y Fm(q)304 1586 y Ft(is)f(connected)j(implies)30 b Fr(U)1255 1601 y Fm(q)1326 1586 y Ft(in)m(tersects)k Fr(@)5 b(S)1874 1601 y Fp([)p Fm(p)p Fp(])1953 1586 y Ft(.)-137 1725 y(Let)31 b(us)f(tak)m(e)h Fr(r)g Fq(2)d Fr(V)38 b Fq(\\)18 b Ft([)p Fr(p)p Ft(])30 b(suc)m(h)i(that)e Fr(q)h Fq(\024)d Fr(r)s Ft(,)j(that)f(is)f(to)h(sa)m (y)38 b(^)-56 b Fr(q)32 b Fq(\032)g Ft(^)-53 b Fr(r)r Ft(.)43 b(Consider)31 b Fr(y)2899 1689 y Fm(r)2895 1750 y Fp(1)2966 1725 y Ft(and)f Fr(y)3205 1689 y Fm(r)3201 1750 y Fp(2)3272 1725 y Ft(asso)s(ciated)g(to)g Fr(r)-236 1865 y Ft(in)c(the)i(same)f(w)m(a)m(y)i(w)m(e)f(did)f(with)g Fr(q)t Ft(.)41 b(Then)29 b Fr(y)1402 1817 y Fm(q)1398 1889 y Fp(1)1467 1865 y Fq(2)35 b Ft(^)-56 b Fr(q)31 b Fq(\032)i Ft(^)-54 b Fr(r)s Ft(.)42 b(On)27 b(the)h(other)f(hand,)i Fr(y)2744 1828 y Fm(r)2740 1889 y Fp(1)2809 1865 y Fq(2)f Fr(S)2963 1880 y Fp([)p Fm(p)p Fp(])3053 1865 y Fq(\\)11 b Fr(@)5 b(B)3260 1880 y Fm(\016)3300 1865 y Ft(\()p Fr(x)p Ft(\))27 b(and)h(b)m(y)g(\()3811 1865 y SDict begin H.S end 3811 1865 a Ft(2)3860 1800 y SDict begin H.R end 3860 1800 a 3860 1865 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.4.2) cvn H.B /ANN pdfmark end 3860 1865 a Ft(\))-236 2004 y Fr(y)-184 1968 y Fm(r)-188 2029 y Fp(1)-119 2004 y Fq(2)35 b Ft(^)-56 b Fr(q)26 b Fq([)c Fr(B)206 2020 y Fm(\032=)p Fp(2)317 2004 y Ft(\()p Fr(@)5 b(S)471 2020 y Fp([)p Fm(p)p Fp(])551 2004 y Ft(\).)43 b(Because)34 b(of)e(\()1181 2004 y SDict begin H.S end 1181 2004 a Ft(1)1230 1940 y SDict begin H.R end 1230 1940 a 1230 2004 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.4.1) cvn H.B /ANN pdfmark end 1230 2004 a Ft(\))h(w)m(e)g(ha)m(v)m(e)h Fr(y)1721 1968 y Fm(r)1717 2029 y Fp(1)1797 2004 y Fr(=)-60 b Fq(2)28 b Fr(B)1954 2020 y Fm(\032=)p Fp(2)2065 2004 y Ft(\()p Fr(@)5 b(S)2219 2020 y Fp([)p Fm(p)p Fp(])2299 2004 y Ft(\))32 b(and)h(so)g Fr(y)2731 1968 y Fm(r)2727 2029 y Fp(1)2795 2004 y Fq(2)j Ft(^)-57 b Fr(q)t Ft(.)44 b(Then)33 b Fr(y)3313 1968 y Fm(r)3309 2029 y Fp(1)3351 2004 y Fr(;)17 b(y)3447 1957 y Fm(q)3443 2028 y Fp(1)3511 2004 y Fq(2)35 b Ft(^)-56 b Fr(q)32 b Fq(\032)g Ft(^)-53 b Fr(r)s Ft(.)-137 2144 y(No)m(w,)30 b(since)f Fr(y)396 2107 y Fm(r)392 2168 y Fp(2)472 2144 y Fr(=)-60 b Fq(2)32 b Ft(^)-53 b Fr(r)31 b Ft(and)36 b(^)-57 b Fr(q)32 b Fq(\032)g Ft(^)-53 b Fr(r)31 b Ft(w)m(e)f(ha)m(v)m(e)f(that)f Fr(y)1689 2107 y Fm(r)1685 2168 y Fp(2)1766 2144 y Fr(=)-61 b Fq(2)35 b Ft(^)-56 b Fr(q)t Ft(.)42 b(Previously)29 b(w)m(e)g(said)f(that)g Fr(y)3034 2096 y Fm(q)3030 2168 y Fp(2)3111 2144 y Fr(=)-61 b Fq(2)28 b Fr(S)3253 2159 y Fp([)p Fm(p)p Fp(])3332 2144 y Ft(,)h(so,)h(applying)-236 2283 y(\()-198 2283 y SDict begin H.S end -198 2283 a Ft(2)-150 2219 y SDict begin H.R end -150 2219 a -150 2283 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.4.2) cvn H.B /ANN pdfmark end -150 2283 a Ft(\))j(\(to)f Fr(r)j Ft(instead)e(of)f Fr(q)t Ft(\))g(w)m(e)i(ha)m(v)m(e)g(that)e Fr(y)1354 2236 y Fm(q)1350 2307 y Fp(2)1431 2283 y Fr(=)-61 b Fq(2)32 b Ft(^)-53 b Fr(r)s Ft(.)43 b(This)33 b(implies)d Fr(y)2236 2236 y Fm(q)2232 2307 y Fp(2)2273 2283 y Fr(;)17 b(y)2369 2247 y Fm(r)2365 2308 y Fp(2)2445 2283 y Fr(=)-60 b Fq(2)32 b Ft(^)-53 b Fr(r)30 b Fq(\033)35 b Ft(^)-56 b Fr(q)t Ft(.)-137 2422 y(Finally)-8 b(,)35 b(since)i Fr(U)528 2437 y Fm(q)600 2422 y Fq(\033)e(f)p Fr(y)814 2375 y Fm(q)810 2447 y Fp(1)851 2422 y Fr(;)17 b(y)947 2375 y Fm(q)943 2447 y Fp(2)984 2422 y Fq(g)36 b Ft(and)g Fr(U)1329 2437 y Fm(r)1401 2422 y Fq(\033)f(f)p Fr(y)1615 2386 y Fm(r)1611 2447 y Fp(1)1652 2422 y Fr(;)17 b(y)1748 2386 y Fm(r)1744 2447 y Fp(2)1785 2422 y Fq(g)36 b Ft(are)g(connected,) j(and)e Fr(S)2778 2437 y Fm(q)2852 2422 y Ft(and)f Fr(S)3105 2437 y Fm(r)3180 2422 y Ft(separate)h(the)f(ball)-236 2562 y Fr(B)-162 2577 y Fm(\027)-119 2562 y Ft(\()p Fr(x)p Ft(\))d(w)m(e)h(deduce)g(that)e Fr(S)785 2577 y Fm(q)845 2562 y Fq(\\)23 b Fr(U)1000 2577 y Fm(r)1070 2562 y Ft(and)33 b Fr(U)1326 2577 y Fm(q)1387 2562 y Fq(\\)22 b Fr(S)1535 2577 y Fm(r)1606 2562 y Ft(are)32 b(not)h(empt)m(y)g(as)f(w)m(an)m (ted.)-137 2701 y(Giv)m(en)37 b Fr(")c(>)h Ft(0,)k(expansivit)m(y)f (and)f(semicon)m(tin)m(uous)h(v)-5 b(ariation)33 b(of)j(lo)s(cal)e (stable)j(and)f(unstable)h(sets)g(allo)m(w)-236 2841 y(us)c(to)e(pro)m(v)m(e)j(that)e(b)m(y)h(means)f(of)f(considering)h Fr(V)54 b Ft(small)29 b(enough)k(w)m(e)g(can)f(ensure)i(that)e(the)h (in)m(tersections)f(lie)-236 2980 y(in)g Fr(B)-48 2995 y Fm(")-12 2980 y Ft(\()p Fr(z)t Ft(\).)3820 3120 y Fj(\003)-137 3259 y Ft(T)-8 b(o)33 b(pro)m(v)m(e)g(Theorem)680 3259 y SDict begin H.S end 680 3259 a Ft(1.2)805 3195 y SDict begin H.R end 805 3195 a 805 3259 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (teo.1.2) cvn H.B /ANN pdfmark end 805 3259 a 32 w Ft(w)m(e)g(shall)e(also)g(mak)m(e)i(use)g(of)e(some)h (prop)s(erties)g(of)g(the)h(fron)m(tier)e(of)h(the)g(sets)i Fr(S)3792 3275 y Fp([)p Fm(p)p Fp(])3871 3259 y Ft(.)-236 3470 y SDict begin H.S end -236 3470 a -236 3470 a SDict begin 16.79996 H.A end -236 3470 a -236 3470 a SDict begin [/View [/XYZ H.V]/Dest (prop.4.1) cvn /DEST pdfmark end -236 3470 a Fy(Prop)s(osition)h(4.1.)42 b Fl(If)p 687 3390 233 4 v 34 w Fr(P)14 b(er)853 3485 y Fm(H)948 3470 y Ft(=)28 b Fr(M)10 b Fl(,)35 b Fr(@)5 b(S)1337 3485 y Fp([)p Fm(p)p Fp(])1439 3470 y Fq(\\)23 b Fr(B)1602 3485 y Fm(\016)1640 3470 y Ft(\()p Fr(x)p Ft(\))35 b Fl(is)f(a)h(top)-5 b(olo)g(gic)g(al)34 b(manifold)f(of)i(dimension)e Fr(n)22 b Fq(\000)h Ft(1)p Fl(.)-236 3710 y Fs(Pr)n(oof.)172 b Ft(Let)47 b Fr(z)57 b Fq(2)52 b Fr(@)5 b(S)804 3726 y Fp([)p Fm(p)p Fp(])916 3710 y Fq(\\)32 b Fr(B)1088 3725 y Fm(\016)1126 3710 y Ft(\()p Fr(x)p Ft(\).)86 b(W)-8 b(e)48 b(c)m(ho)s(ose)f Fr(")52 b(>)g Ft(0)46 b(suc)m(h)i(that)f Fr(B)2731 3725 y Fm(")2768 3710 y Ft(\()p Fr(z)t Ft(\))52 b Fq(\032)h Fr(B)3149 3725 y Fm(\016)3187 3710 y Ft(\()p Fr(x)p Ft(\))47 b(and)f(let)h Fr(V)68 b Ft(a)-236 3850 y(neigh)m(b)s(orho)s(o)s(d)28 b(of)h Fr(z)35 b Ft(satisfying)28 b(that)i(if)e Fr(q)t(;)17 b(r)30 b Fq(2)e Fr(V)38 b Fq(\\)16 b Ft([)p Fr(p)p Ft(])30 b(then)g Fr(U)2141 3865 y Fm(q)2195 3850 y Fq(\\)16 b Fr(S)2337 3865 y Fm(r)2392 3850 y Fq(\\)g Fr(B)2548 3865 y Fm(")2585 3850 y Ft(\()p Fr(z)t Ft(\))28 b Fq(6)p Ft(=)g Fq(;)h Ft(as)h(giv)m(en)g(in)f(Lemma)3746 3850 y SDict begin H.S end 3746 3850 a Ft(4.6)3871 3786 y SDict begin H.R end 3871 3786 a 3871 3850 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lema.4.6) cvn H.B /ANN pdfmark end 3871 3850 a Ft(.)-236 3989 y(Also,)j(for)g(ev)m(ery)i Fr(q)e Fq(2)c Fr(V)44 b Fq(\\)22 b Ft([)p Fr(p)p Ft(])33 b(w)m(e)g(can)g(ha)m(v)m(e)h Fr(U)1524 4004 y Fm(q)1585 3989 y Fq(\\)22 b Fr(@)5 b(S)1789 4005 y Fp([)p Fm(p)p Fp(])1891 3989 y Fq(\\)23 b Fr(B)2054 4004 y Fm(")2091 3989 y Ft(\()p Fr(z)t Ft(\))28 b Fq(6)p Ft(=)f Fq(;)33 b Ft(again)e(b)m(y)i(Lemma)3173 3989 y SDict begin H.S end 3173 3989 a Ft(4.6)3298 3925 y SDict begin H.R end 3298 3925 a 3298 3989 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lema.4.6) cvn H.B /ANN pdfmark end 3298 3989 a Ft(.)-137 4129 y(Pic)m(k)h Fr(q)d Fq(2)d Fr(V)44 b Fq(\\)22 b Ft([)p Fr(p)p Ft(])33 b(and)g(de\014ne)h Fr(h)1103 4144 y Fm(q)1152 4129 y Ft(:)f Fr(S)1272 4144 y Fm(q)1332 4129 y Fq(\\)p 1421 4049 79 4 v 23 w Fr(V)49 b Fq(!)27 b Fr(@)5 b(S)1770 4144 y Fp([)p Fm(p)p Fp(])1872 4129 y Fq(\\)p 1961 4042 237 4 v 23 w Fr(B)2035 4144 y Fm(")2072 4129 y Ft(\()p Fr(z)t Ft(\))33 b(giv)m(en)g(b)m(y)1330 4410 y Fr(h)1386 4425 y Fm(q)1424 4410 y Ft(\()p Fr(y)t Ft(\))27 b(=)50 b(lim)1682 4470 y Fm(q)1714 4478 y Fi(n)1756 4470 y Fk(!)p Fm(y)1881 4410 y Fr(U)1947 4425 y Fm(q)1979 4433 y Fi(n)2047 4410 y Fq(\\)23 b Fr(@)5 b(S)2252 4425 y Fp([)p Fm(p)p Fp(])-236 4597 y Ft(whic)m(h)27 b(is)f(w)m(ell)f(de\014ned)j(thanks)g (to)e(expansivit)m(y)h(and)f(semicon)m(tin)m(uous)h(v)-5 b(ariation)24 b(of)i(lo)s(cal)e(stable)i(and)h(unsta-)-236 4736 y(ble)33 b(sets)i(\(Lemma)501 4736 y SDict begin H.S end 501 4736 a Ft(3.2)626 4672 y SDict begin H.R end 626 4672 a 626 4736 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lema.3.2) cvn H.B /ANN pdfmark end 626 4736 a Ft(\))e(together)h (with)f(the)h(fact)g(that)f Fr(@)5 b(S)1997 4752 y Fp([)p Fm(p)p Fp(])2106 4736 y Fq(\032)30 b Fr(W)2319 4700 y Fm(s)2305 4761 y(\015)2356 4736 y Ft(\()p Fr(z)t Ft(\))k(b)s(ecause)h (of)e(Lemma)3338 4736 y SDict begin H.S end 3338 4736 a Ft(4.5)3463 4672 y SDict begin H.R end 3463 4672 a 3463 4736 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lema.4.5) cvn H.B /ANN pdfmark end 3463 4736 a Ft(.)46 b(The)35 b(fact)-236 4876 y(that)28 b(there)h(is)f(a)g(sequence)j Fr(q)830 4891 y Fm(n)905 4876 y Fq(2)d Ft([)p Fr(p)p Ft(])g Fq(!)f Fr(y)k Ft(is)d(a)g(consequence)k(of)27 b(Lemma)2504 4876 y SDict begin H.S end 2504 4876 a Ft(4.3)2629 4811 y SDict begin H.R end 2629 4811 a 2629 4876 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lema.4.3) cvn H.B /ANN pdfmark end 2629 4876 a 28 w Ft(and)i(the)f(fact)h(that)p 3402 4796 233 4 v 28 w Fr(P)14 b(er)3568 4891 y Fm(H)3662 4876 y Ft(=)28 b Fr(M)10 b Ft(.)-137 5015 y(The)34 b(same)f(argumen)m (t)g(implies)e(that)h Fr(h)1345 5030 y Fm(q)1417 5015 y Ft(is)g(con)m(tin)m(uous)i(and)f(injectiv)m(e.)45 b(Moreo)m(v)m(er,) 35 b(since)e(the)h(domain)d(is)-236 5155 y(compact,)h Fr(h)237 5170 y Fm(q)308 5155 y Ft(is)g(a)g(homeomorphism)e(o)m(v)m(er) k(its)e(image.)-137 5294 y(Again,)24 b(b)m(y)g(Lemma)631 5294 y SDict begin H.S end 631 5294 a Ft(4.6)756 5230 y SDict begin H.R end 756 5230 a 756 5294 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lema.4.6) cvn H.B /ANN pdfmark end 756 5294 a 23 w Ft(,)g(w)m(e)g(can)f(tak)m(e)h Fr(V)1413 5258 y Fk(0)1459 5294 y Ft(and)f Fr(")1685 5258 y Fk(0)1736 5294 y Fr(>)k Ft(0)c(suc)m(h)h(that)p 2322 5207 259 4 v 22 w Fr(B)2396 5309 y Fm(")2429 5290 y Ff(0)2456 5294 y Ft(\()p Fr(z)t Ft(\))k Fq(\032)g Fr(V)44 b Ft(and)23 b(that)g(for)f Fr(q)t(;)17 b(r)30 b Fq(2)e Ft([)p Fr(p)p Ft(])r Fq(\\)r Fr(V)3847 5258 y Fk(0)3871 5294 y Ft(,)-236 5434 y Fr(U)-170 5449 y Fm(q)-106 5434 y Fq(\\)d Fr(S)45 5449 y Fm(r)109 5434 y Fq(\\)h Fr(B)275 5449 y Fm(")308 5430 y Ff(0)334 5434 y Ft(\()p Fr(z)t Ft(\))37 b Fq(6)p Ft(=)f Fq(;)p Ft(.)58 b(Analogously)-8 b(,)37 b(w)m(e)i(ha)m(v)m(e)g (that)e(for)g(ev)m(ery)i Fr(q)h Fq(2)d Fr(V)2596 5397 y Fk(0)2645 5434 y Fq(\\)26 b Ft([)p Fr(p)p Ft(],)39 b Fr(U)2972 5449 y Fm(q)3036 5434 y Fq(\\)26 b Fr(@)5 b(S)3244 5449 y Fp([)p Fm(p)p Fp(])3349 5434 y Fq(\\)p 3441 5347 V 26 w Fr(B)3515 5449 y Fm(")3548 5430 y Ff(0)3574 5434 y Ft(\()p Fr(z)t Ft(\))37 b Fq(6)p Ft(=)f Fq(;)-236 5573 y Ft(v)m(eri\014es.)-137 5712 y(If)43 b(w)m(e)h(\014x)f Fr(q)48 b Fq(2)d Fr(V)554 5676 y Fk(0)606 5712 y Fq(\\)30 b Ft([)p Fr(p)p Ft(])42 b(\(Lemmas)1282 5712 y SDict begin H.S end 1282 5712 a Ft(4.3)1406 5648 y SDict begin H.R end 1406 5648 a 1406 5712 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lema.4.3) cvn H.B /ANN pdfmark end 1406 5712 a 43 w Ft(and)1649 5712 y SDict begin H.S end 1649 5712 a Ft(4.5)1773 5648 y SDict begin H.R end 1773 5648 a 1773 5712 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lema.4.5) cvn H.B /ANN pdfmark end 1773 5712 a 43 w Ft(and)g(ensures)j(the)e (existence)h(of)e(suc)m(h)i Fr(q)t Ft(\))f(w)m(e)g(will)d(b)s(e)-236 5852 y(able)i(to)g(pro)m(v)m(e)i(that)e(for)g(all)f Fr(w)47 b Fq(2)e Fr(@)5 b(S)1252 5867 y Fp([)p Fm(p)p Fp(])1361 5852 y Fq(\\)30 b Fr(V)1535 5816 y Fk(0)1601 5852 y Ft(exists)43 b Fr(y)48 b Fq(2)d Fr(S)2147 5867 y Fm(q)2214 5852 y Fq(\\)30 b Fr(V)64 b Ft(suc)m(h)44 b(that)e Fr(h)2938 5867 y Fm(q)2977 5852 y Ft(\()p Fr(y)t Ft(\))h(=)i Fr(w)s Ft(.)73 b(This)43 b(holds)p eop end end %%Page: 14 14 TeXDict begin HPSdict begin 14 13 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a -245 -8 a SDict begin H.S end -245 -8 a -245 -8 a 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Ft(4.1)514 545 y SDict begin H.R end 514 545 a 514 609 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (prop.4.1) cvn H.B /ANN pdfmark end 514 609 a 33 w Ft(ensures)36 b(that)d Fr(@)5 b(S)1219 625 y Fp([)p Fm(p)p Fp(])1322 609 y Fq(\\)23 b Fr(B)1485 624 y Fm(\016)1523 609 y Ft(\()p Fr(x)p Ft(\))34 b(is)f(a)g(top)s(ological)d(manifold)h (of)i(dimension)f Fr(n)23 b Fq(\000)g Ft(1.)46 b(Then,)-236 749 y(Lemma)120 749 y SDict begin H.S end 120 749 a Ft(4.8)244 685 y SDict begin H.R end 244 685 a 244 749 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lema.4.8) cvn H.B /ANN pdfmark end 244 749 a 41 w Ft(and)40 b(a)g(result)g(in)g([)999 749 y SDict begin H.S end 999 749 a Ft(HW)1172 681 y SDict begin H.R end 1172 681 a 1172 749 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.HW) cvn H.B /ANN pdfmark end 1172 749 a 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Ft(HW)-36 1633 y SDict begin H.R end -36 1633 a -36 1701 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.HW) cvn H.B /ANN pdfmark end -36 1701 a Ft(])e(c)m(hapter)h(I)s(I)s(I,)f(section)g(3\))f(w)m(e)i (kno)m(w)g(that)f(dim)m(top)o(\()p Fr(@)5 b(S)h Ft(\))33 b Fq(\024)h Fr(n)25 b Fq(\000)g Ft(2.)53 b(So,)36 b Fr(@)5 b(S)43 b Ft(can)36 b(not)g(separate)g Fr(B)3728 1716 y Fm(\016)3766 1701 y Ft(\()p Fr(x)p Ft(\))-236 1841 y(b)s(ecause)h(it)d(should)h(ha)m(v)m(e)i(dimension)d(at)h(least)g Fr(n)24 b Fq(\000)h Ft(1)35 b(\(see)i([)2079 1841 y SDict begin H.S end 2079 1841 a Ft(HW)2252 1773 y SDict begin H.R end 2252 1773 a 2252 1841 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.HW) cvn H.B /ANN pdfmark end 2252 1841 a Ft(])f(c)m(hapter)g(IV,)g(section)f(5\).)52 b(This)36 b(leads)f(us)-236 1980 y(to)d(a)g(con)m(tradiction.)-236 2150 y Fs(Pr)n(oof)38 b(of)g(Lemma)639 2150 y SDict begin H.S end 639 2150 a Fs(4.8)781 2086 y SDict begin H.R end 781 2086 a 781 2150 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lema.4.8) cvn H.B /ANN pdfmark end 781 2150 a Fs(.)-137 2289 y Ft(Let)h Fr(")d(>)h Ft(0)g(and)h Fr(z)k Fq(2)37 b Fr(@)5 b(S)826 2305 y Fp([)p Fm(p)p Fp(])906 2289 y Ft(.)60 b(By)38 b(Lemma)1505 2289 y SDict begin H.S end 1505 2289 a Ft(4.6)1629 2225 y SDict begin H.R end 1629 2225 a 1629 2289 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lema.4.6) cvn H.B /ANN pdfmark end 1629 2289 a Ft(,)i(there)e(exist)g Fr(q)j Fq(2)c Ft([)p Fr(p)p Ft(])h(suc)m(h)i(that)e Fq(f)p Fr(a)p Fq(g)e Ft(=)h Fr(U)3323 2304 y Fm(q)3387 2289 y Fq(\\)26 b Fr(@)5 b(S)3595 2305 y Fp([)p Fm(p)p Fp(])3713 2289 y Ft(is)37 b(in)-236 2429 y Fr(B)-162 2444 y Fm(")-125 2429 y Ft(\()p Fr(z)t Ft(\).)45 b(Theorem)484 2429 y SDict begin H.S end 484 2429 a Ft(1.1)609 2364 y SDict begin H.R end 609 2364 a 609 2429 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (teo.1.1) cvn H.B /ANN pdfmark end 609 2429 a 33 w Ft(implies)30 b(that)j Fr(q)k Ft(has)c(a)g(neigh)m(b)s(orho)s(o)s(d)f(with)g(lo)s(cal)f(pro)s (duct)i(structure,)i(b)m(y)f(iterating)-236 2568 y(this)f(neigh)m(b)s (orho)s(o)s(d)g(to)g(the)i(past,)f(w)m(e)h(obtain)e(lo)s(cal)f(pro)s (duct)i(structure)h(o)m(v)m(er)g(a)e(neigh)m(b)s(orho)s(o)s(d)g(of)h Fr(a)p Ft(,)g(so,)g Fr(a)-236 2708 y Ft(m)m(ust)e(b)s(elong)g(to)g Fr(int)p Ft(\()p Fr(S)6 b Ft(\))28 b(=)g Fr(int)p Ft(\()p 1000 2612 306 4 v Fr(P)14 b(er)1169 2666 y Fm(n)p Fk(\000)p Fp(1)1166 2735 y Fm(H)1328 2708 y Fq(\\)23 b Fr(B)1491 2723 y Fm(\016)1529 2708 y Ft(\()p Fr(x)p Ft(\)\))33 b(and)f(the)h(lemma)e(is)h(pro)m(v)m(ed.)3820 2847 y Fj(\003)-236 2905 y SDict begin H.S end -236 2905 a -236 2905 a SDict begin 16.79996 H.A end -236 2905 a -236 2905 a SDict begin [/View [/XYZ H.V]/Dest (subsection.4.3) cvn /DEST pdfmark end -236 2905 a 140 x Ft(4.3.)48 b Fy(Uniform)34 b(lo)s(cal)g(pro)s(duct)h(structure.)48 b Ft(W)-8 b(e)31 b(shall)e(pro)m(v)m(e)j(Theorem)2680 3045 y SDict begin H.S end 2680 3045 a Ft(1.3)2804 2981 y SDict begin H.R end 2804 2981 a 2804 3045 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (teo.1.3) cvn H.B /ANN pdfmark end 2804 3045 a 31 w Ft(in)d(this)h(section.)43 b(By)31 b(Theo-)-236 3184 y(rem)-47 3184 y SDict begin H.S end -47 3184 a Ft(1.1)78 3120 y SDict begin H.R end 78 3120 a 78 3184 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (teo.1.1) cvn H.B /ANN pdfmark end 78 3184 a 26 w Ft(w)m(e)d(kno)m(w)g(that)e(there)h(is) f(an)h(op)s(en)g(and)f(dense)i(set)g(whose)g(p)s(oin)m(ts)e(admit)f(a)h (lo)s(cal)e(pro)s(duct)j(structure.)-236 3324 y(And)33 b(b)m(y)g(Theorem)525 3324 y SDict begin H.S end 525 3324 a Ft(1.2)650 3259 y SDict begin H.R end 650 3259 a 650 3324 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (teo.1.2) cvn H.B /ANN pdfmark end 650 3324 a 32 w Ft(w)m(e)h(conclude,)f(since)g Fr(P)14 b(er)1662 3282 y Fm(n)p Fk(\000)p Fp(1)1659 3351 y Fm(H)1826 3324 y Fq(6)p Ft(=)28 b Fq(;)p Ft(,)k(that)h Fr(P)14 b(er)2417 3339 y Fm(H)2511 3324 y Ft(=)28 b Fr(P)14 b(er)2784 3282 y Fm(n)p Fk(\000)p Fp(1)2781 3351 y Fm(H)2920 3324 y Ft(.)-137 3463 y(Let)38 b Fq(S)45 b Ft(b)s(e)37 b(the)g(set)h(of)f(singularities)e(of)h Fr(f)11 b Ft(,)38 b(that)f(is)g(to)g(sa)m(y)-8 b(,)39 b(the)e(p)s(oin)m(ts)g(whic)m(h)h (do)f(not)g(admit)e(an)m(y)j(lo)s(cal)-236 3603 y(pro)s(duct)33 b(structure.)44 b(T)-8 b(o)33 b(pro)m(v)m(e)h(Theorem)1404 3603 y SDict begin H.S end 1404 3603 a Ft(1.3)1528 3538 y SDict begin H.R end 1528 3538 a 1528 3603 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (teo.1.3) cvn H.B /ANN pdfmark end 1528 3603 a 33 w Ft(w)m(e)f(m)m(ust)g(pro)m(v)m(e)h(that)e Fq(S)41 b Ft(is)32 b(an)g(empt)m(y)h(set.)-137 3742 y(With)38 b(the)h(results)g(pro)m(v)m(ed)h(in)1054 3742 y SDict begin H.S end 1054 3742 a Ft(4.1)1178 3678 y SDict begin H.R end 1178 3678 a 1178 3742 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (subsection.4.1) cvn H.B /ANN pdfmark end 1178 3742 a 39 w Ft(w)m(e)f(obtain)f(the)h(follo)m(wing)c(consequence)42 b(whic)m(h)d(allo)m(ws)e(us)i(to)f(study)-236 3882 y(the)29 b(set)h(of)e(singularities)e(in)i(co)s(dimension)g(one)h(case.)43 b(The)30 b(next)g(prop)s(osition)d(giv)m(es)i(a)f(sort)h(of)g(lo)s(cal) d(pro)s(duct)-236 4021 y(structure)34 b(in)d(the)i(sets)h Fr(S)714 4037 y Fp([)p Fm(p)p Fp(])826 4021 y Ft(whic)m(h)f(will)d(b)s (e)j(de\014ned)h(prop)s(erly)e(in)g(this)g(statemen)m(t.)-236 4224 y SDict begin H.S end -236 4224 a -236 4224 a SDict begin 16.79996 H.A end -236 4224 a -236 4224 a SDict begin [/View [/XYZ H.V]/Dest (prop.4.2) cvn /DEST pdfmark end -236 4224 a Fy(Prop)s(osition)38 b(4.2.)43 b Fl(L)-5 b(et)38 b Fr(x)32 b Fq(2)h(S)7 b Fl(.)52 b(Then,)37 b(for)g(every)g Fr(z)g Fq(2)32 b Fr(@)5 b(S)2096 4240 y Fp([)p Fm(p)p Fp(])2199 4224 y Fq(\\)25 b Fr(B)2364 4239 y Fm(\016)2402 4224 y Ft(\()p Fr(x)p Ft(\))37 b Fl(ther)-5 b(e)37 b(exists)g Fr(h)32 b Ft(:)g Fr(I)g Fq(\002)24 b Fr(I)3458 4188 y Fm(n)p Fk(\000)p Fp(1)3627 4224 y Fq(!)32 b Fr(S)3819 4240 y Fp([)p Fm(p)p Fp(])-236 4364 y Fl(\()p Fr(I)41 b Ft(=)33 b([0)p Fr(;)17 b Ft(1])p Fl(\))37 b(home)-5 b(omorphism)36 b(over)i(its)g(image,)f(wher)-5 b(e)38 b Fr(h)p Ft(\()p Fq(f)p Fr(a)p Fq(g)24 b(\002)h Fr(I)2374 4327 y Fm(n)2421 4364 y Ft(\))38 b Fl(is)f(c)-5 b(ontaine)g(d)37 b(in)h(a)g(lo)-5 b(c)g(al)37 b(stable)h(set,)-236 4503 y Fr(h)p Ft(\()p Fr(I)31 b Fq(\002)25 b(f)p Fr(b)p Fq(g)p Ft(\))37 b Fl(is)g(c)-5 b(ontaine)g(d)36 b(in)h(a)f(lo)-5 b(c)g(al)37 b(unstable)g(set)g(and)f(the)h(image)f(of)h Fr(h)g Fl(is)g(a)g(neighb)-5 b(orho)g(o)g(d)35 b(of)i Fr(z)42 b Fl(r)-5 b(elative)-236 4643 y(to)35 b Fr(S)-59 4658 y Fp([)p Fm(p)p Fp(])20 4643 y Fl(.)-236 4876 y Fs(Pr)n(oof.)84 b Ft(By)28 b(Lemma)682 4876 y SDict begin H.S end 682 4876 a Ft(4.6)807 4811 y SDict begin H.R end 807 4811 a 807 4876 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lema.4.6) cvn H.B /ANN pdfmark end 807 4876 a 28 w Ft(there)g(exists)h Fr(V)49 b Fq(\032)28 b Fr(B)1629 4891 y Fm(\016)1667 4876 y Ft(\()p Fr(x)p Ft(\))g(neigh)m(b)s(orho)s(o) s(d)f(of)g Fr(z)33 b Ft(in)27 b Fr(M)38 b Ft(suc)m(h)30 b(that)d(if)g Fr(q)t(;)17 b(r)30 b Fq(2)e Ft([)p Fr(p)p Ft(])12 b Fq(\\)g Fr(V)-236 5015 y Ft(then)33 b Fr(S)46 5030 y Fm(q)106 5015 y Fq(\\)23 b Fr(U)261 5030 y Fm(r)327 5015 y Fq(6)p Ft(=)k Fq(;)32 b Ft(and)h Fr(U)768 5030 y Fm(q)829 5015 y Fq(\\)22 b Fr(@)5 b(S)1033 5031 y Fp([)p Fm(p)p Fp(])1141 5015 y Fq(6)p Ft(=)27 b Fq(;)p Ft(.)-137 5155 y(Let)42 b Fr(D)128 5170 y Fm(z)211 5155 y Fq(\032)h Fr(@)5 b(S)447 5170 y Fp([)p Fm(p)p Fp(])555 5155 y Fq(\\)29 b Fr(V)63 b Ft(homeomorphic)39 b(to)j Fr(I)1614 5118 y Fm(n)p Fk(\000)p Fp(1)1792 5155 y Ft(\(see)h(Prop)s(osition)2530 5155 y SDict begin H.S end 2530 5155 a Ft(4.1)2655 5090 y SDict begin H.R end 2655 5090 a 2655 5155 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (prop.4.1) cvn H.B /ANN pdfmark end 2655 5155 a Ft(\))e(suc)m(h)i(that)f Fr(z)k Ft(b)s(elongs)41 b(to)g(the)-236 5294 y(in)m(terior)31 b(of)h Fr(D)300 5309 y Fm(z)372 5294 y Ft(relativ)m(e)g(to)g Fr(@)5 b(S)954 5310 y Fp([)p Fm(p)p Fp(])1034 5294 y Ft(.)-137 5434 y(Let)33 b Fr(V)117 5397 y Fk(0)168 5434 y Fq(\032)28 b Fr(V)54 b Ft(neigh)m(b)s(orho)s(o)s(d)31 b(of)h Fr(z)37 b Ft(suc)m(h)d(that)f(if)e Fr(q)h Fq(2)c Ft([)p Fr(p)p Ft(])22 b Fq(\\)g Fr(V)2171 5397 y Fk(0)2227 5434 y Ft(then)33 b Fr(S)2509 5449 y Fm(q)2569 5434 y Fq(\\)22 b Fr(U)2723 5449 y Fm(r)2784 5434 y Fq(\\)g Fr(V)49 b Fq(6)p Ft(=)28 b Fq(;)k Ft(and)h Fr(U)3420 5449 y Fm(q)3480 5434 y Fq(\\)23 b Fr(D)3650 5449 y Fm(z)3717 5434 y Fq(6)p Ft(=)28 b Fq(;)p Ft(.)-137 5573 y(Let)42 b Fr(q)k Fq(2)e Fr(V)324 5537 y Fk(0)376 5573 y Fq(\\)28 b Ft([)p Fr(p)p Ft(])42 b(and)f(w)m(e)h(de\014ne)h Fr(h)g Ft(:)g Fr(U)1491 5588 y Fm(q)1557 5573 y Fq(\\)p 1652 5493 79 4 v 29 w Fr(V)49 b Fq(\\)29 b Fr(S)1913 5588 y Fp([)p Fm(p)p Fp(])2020 5573 y Fq(\002)g Fr(D)2207 5588 y Fm(z)2289 5573 y Fq(!)42 b Fr(S)2491 5588 y Fp([)p Fm(p)p Fp(])2611 5573 y Ft(in)f(suc)m(h)i(a)e (w)m(a)m(y)i(that)e Fr(h)p Ft(\()p Fr(y)t(;)17 b(w)s Ft(\))41 b(=)-236 5712 y Fr(W)-130 5676 y Fm(s)-144 5737 y(")-94 5712 y Ft(\()p Fr(y)t Ft(\))30 b Fq(\\)i Fr(W)268 5676 y Fm(u)254 5737 y(")313 5712 y Ft(\()p Fr(w)s Ft(\))45 b(is)g(v)m(eri\014ed.)84 b(By)46 b(the)g(c)m(hoice)h(of)e Fr(V)21 b Ft(,)50 b(appro)m(ximating)43 b(with)j(top)s(ologically)41 b(h)m(yp)s(erb)s(olic)-236 5852 y(p)s(erio)s(dic)c(p)s(oin)m(ts)h(and)h (making)e(use)j(of)e(expansivit)m(y)i(and)e(semicon)m(tin)m(uous)h(v)-5 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b(dense\))h(w)m(e)g(ha)m(v)m(e)h(that)-236 2919 y Fr(B)-162 2934 y Fm(\016)-124 2919 y Ft(\()p Fr(x)p Ft(\))c Fq(\032)167 2845 y Fh(S)250 2871 y Fm(k)250 2948 y(i)p Fp(=1)385 2919 y Fr(S)445 2935 y Fp([)p Fm(p)501 2945 y Fi(i)526 2935 y Fp(])550 2919 y Ft(.)67 b(Prop)s(osition)1177 2919 y SDict begin H.S end 1177 2919 a Ft(4.2)1301 2855 y SDict begin H.R end 1301 2855 a 1301 2919 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (prop.4.2) cvn H.B /ANN pdfmark end 1301 2919 a 41 w Ft(implies)38 b(that)i(in)f(the)i(in)m(terior)e (of)h Fr(S)2728 2935 y Fp([)p Fm(p)2784 2945 y Fi(i)2810 2935 y Fp(])2874 2919 y Ft(there)h(is)f(a)g(lo)s(cal)e(pro)s(duct)-236 3059 y(structure)29 b(\(ma)m(yb)s(e)e(b)m(y)h(considering)f Fr(\016)32 b Ft(smaller\))25 b(so)j(singularities)d(m)m(ust)i(lie)f(in) 2723 2984 y Fh(S)2806 3010 y Fm(k)2806 3088 y(i)p Fp(=1)2941 3059 y Fr(@)5 b(S)3057 3074 y Fp([)p Fm(p)3113 3084 y Fi(i)3139 3074 y Fp(])3163 3059 y Ft(.)42 b(Lemma)3575 3059 y SDict begin H.S end 3575 3059 a Ft(4.2)3700 2995 y SDict begin H.R end 3700 2995 a 3700 3059 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lema.4.2) cvn H.B /ANN pdfmark end 3700 3059 a 27 w Ft(no)m(w)-236 3198 y(implies)30 b(that)i(singularities)e(of)i Fr(B)1034 3213 y Fm(\016)1072 3198 y Ft(\()p Fr(x)p Ft(\))h(b)s(elong)f(to)g(the)h(lo)s(cal)d(stable) j(set)g(of)f Fr(x)p Ft(.)-137 3338 y(Expansivit)m(y)37 b(implies)c(that)j(Ly)m(apuno)m(v)h(stable)f(p)s(oin)m(ts)f(are)h (asymptotically)d(stable.)53 b(Otherwise,)37 b(p)s(oin)m(ts)-236 3477 y Fr(y)t(;)17 b(w)35 b Ft(suc)m(h)h(that)e(dist\()p Fr(f)655 3441 y Fm(n)702 3477 y Ft(\()p Fr(y)t Ft(\))p Fr(;)17 b(f)933 3441 y Fm(n)978 3477 y Ft(\()p Fr(w)s Ft(\)\))30 b Fq(\024)h Fr(")f Fq(\024)h Fr(\013)k Ft(\()p Fr(\013)g Ft(expansivit)m(y)g(constan)m(t\))h(and)e(suc)m(h)i(that)e(a) g(subsequence)-236 3617 y Fr(n)-178 3632 y Fm(j)-114 3617 y Fq(!)28 b Ft(+)p Fq(1)k Ft(with)g(dist)o(\()p Fr(f)698 3581 y Fm(n)741 3591 y Fi(j)778 3617 y Ft(\()p Fr(y)t Ft(\))p Fr(;)17 b(f)1009 3581 y Fm(n)1052 3591 y Fi(j)1086 3617 y Ft(\()p Fr(w)s Ft(\)\))27 b Fq(\025)i Fr(\016)36 b Ft(exist.)44 b(T)-8 b(aking)32 b(limit)d(p)s(oin)m(ts)j(w) m(e)i(con)m(tradict)e(expansivit)m(y)-8 b(.)-137 3756 y(Since)32 b Fq(S)40 b Ft(is)31 b(compact)h(and)f(ev)m(ery)j(p)s(oin)m (t)d(is)g(asymptotically)f(stable)h(for)h Fr(f)11 b Ft(,)31 b(w)m(e)i(conclude)f(that)g Fq(S)39 b Ft(m)m(ust)32 b(b)s(e)-236 3896 y(\014nite.)3820 4035 y Fj(\003)-137 4175 y Ft(In)40 b(the)f(follo)m(wing)e(Lemma)g(w)m(e)k(will)36 b(sho)m(w)41 b(that)d(there)i(are)f(no)g(isolated)f(singularities)f(if)h(dim)n(\()p Fr(M)10 b Ft(\))40 b Fq(\025)f Ft(3.)-236 4314 y(Observ)m(e)31 b(that)e(in)g(surfaces,)i(pseudoAnoso)m(v)g(maps)e(ha)m(v)m(e)i(this)e (kind)g(of)g(singular)f(p)s(oin)m(ts.)42 b(The)30 b(k)m(ey)h(fact)e (here)-236 4454 y(is)g(ho)m(w)h(the)g(semilo)s(cal)d(pro)s(duct)j 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y(l)q(oc)2374 4733 y Ft(\()p Fr(x)p Ft(\))20 b Fq(n)g(f)p Fr(x)p Fq(g)31 b Ft(is)g(connected)i(and)f(therefore)-236 4872 y(there)h(is)f(no)g (place)g(for)g(a)g(third)g(semilo)s(cal)d(pro)s(duct)k(structure.)45 b(This)32 b(will)e(let)i(us)h(pro)m(v)m(e)h(that)e Fr(x)h Ft(has)g(a)f(lo)s(cal)-236 5011 y(pro)s(duct)h(structure.)-236 5207 y SDict begin H.S end -236 5207 a -236 5207 a SDict begin 16.79996 H.A end -236 5207 a -236 5207 a SDict begin [/View [/XYZ H.V]/Dest (lema.4.10) cvn /DEST pdfmark end -236 5207 a Fy(Lemma)k(4.10.)42 b Fl(If)34 b Ft(dim)o(\()p Fr(M)10 b Ft(\))28 b Fq(\025)g Ft(3)35 b Fl(and)p 1284 5112 306 4 v 34 w Fr(P)14 b(er)1453 5166 y Fm(n)p Fk(\000)p Fp(1)1450 5235 y Fm(H)1617 5207 y Ft(=)28 b Fr(M)45 b Fl(then,)35 b(no)f(isolate)-5 b(d)35 b(singularities)f(exist.)-236 5434 y Fs(Pr)n(oof.)84 b Ft(By)47 b(con)m(tradiction,)i(supp)s(ose)e Fr(x)k Fq(2)g Fr(M)57 b Ft(is)46 b(an)g(isolated)f(singularit)m(y)-8 b(.)82 b(Let)47 b Fr(\027)q(;)17 b(\016)54 b(>)d Ft(0)46 b(b)s(e)g(as)g(in)-236 5573 y(Prop)s(osition)291 5573 y SDict begin H.S end 291 5573 a Ft(4.1)415 5509 y SDict begin H.R end 415 5509 a 415 5573 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (prop.4.1) cvn H.B /ANN pdfmark end 415 5573 a 35 w Ft(and)35 b(suc)m(h)h(that)e Fr(B)1151 5588 y Fm(\027)1194 5573 y Ft(\()p Fr(x)p Ft(\))24 b Fq(\\)g(S)39 b Ft(=)31 b Fq(f)p Fr(x)p Fq(g)p Ft(.)49 b(Fix)34 b([)p Fr(p)p Ft(])h(a)f(maximal)d(c)m(hain)k(accum)m(ulating)e (in)h Fr(x)g Ft(and)-236 5712 y(let)g Fr(T)46 b Ft(=)33 b Fr(cc)203 5727 y Fm(x)246 5712 y Ft(\()p Fr(@)5 b(S)400 5728 y Fp([)p Fm(p)p Fp(])504 5712 y Fq(\\)25 b Fr(B)669 5727 y Fm(\016)707 5712 y Ft(\()p Fr(x)p Ft(\)\).)51 b(W)-8 b(e)36 b(kno)m(w)g(b)m(y)h(Prop)s(osition)2048 5712 y SDict begin H.S end 2048 5712 a Ft(4.1)2173 5648 y SDict begin H.R end 2173 5648 a 2173 5712 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest 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Fe(Er)l(go)l(dic)k(The)l(ory)g(and)f(Dy-)-107 3912 y(namic)l(al)d (Systems)p Fx(,)c Fd(22)g Fx(\(2002\),)f(601{632.)-137 4136 y Fw(Imerl,)31 b(F)-10 b(a)n(cul)k(t)g(ad)32 b(de)g(Ingenier)1045 4129 y(\023)1055 4136 y(\020a,)f(Universid)n(ad)g(de)h(la)e(Rep)2142 4129 y(\023)2139 4136 y(ublica,)i(Ur)n(ugua)-6 b(y)-137 4256 y Fe(E-mail)31 b(addr)l(ess)7 b Fx(:)38 b Fa(aartigue@fing.ed)o (u.u)o(y)-137 4459 y Fw(Cma)-6 b(t,)32 b(F)-10 b(a)n(cul)k(t)g(ad)32 b(de)g(Ciencias,)f(Universid)n(ad)g(de)h(la)f(Rep)2048 4452 y(\023)2045 4459 y(ublica,)g(Ur)n(ugua)-6 b(y)-137 4578 y Fe(E-mail)31 b(addr)l(ess)7 b Fx(:)38 b Fa(joaquin@cmat.edu)o (.uy)-137 4782 y Fw(Imerl,)31 b(F)-10 b(a)n(cul)k(t)g(ad)32 b(de)g(Ingenier)1045 4775 y(\023)1055 4782 y(\020a,)f(Universid)n(ad)g (de)h(la)e(Rep)2142 4775 y(\023)2139 4782 y(ublica,)i(Ur)n(ugua)-6 b(y)-137 4901 y Fe(E-mail)31 b(addr)l(ess)7 b Fx(:)38 b Fa(rpotrie@cmat.edu)o(.uy)p eop end end %%Trailer end userdict /end-hook known{end-hook}if %%EOF