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b(to)e(b)s(e)g Fl(C)1198 515 y Fj(r)1266 548 y Fn(i\033)g Fl(X)37 b Fn(is)30 b(a)h Fl(C)1725 515 y Fj(r)1792 548 y Fn(-)f(v)m(ector)i(\034eld)f(on)f Fl(M)10 b Fn(.)722 774 y Fm(Def:)54 b Fl(M)5 b(;)15 b(N)48 b Fn(as)37 b(b)s(efore.)62 b(T)-8 b(w)m(o)39 b Fl(C)1939 741 y Fj(r)2014 774 y Fn(-)e(\035o)m(ws)h Fl(\036)2366 788 y Fj(t)2433 774 y Fi(:)g Fl(M)47 b Fk(!)37 b Fl(M)5 b(;)15 b( )2951 788 y Fj(t)3018 774 y Fi(:)38 b Fl(N)47 b Fk(!)37 b Fl(N)47 b Fn(are)581 887 y(called)39 b Fl(C)917 854 y Fg(0)995 887 y Fh(-)g(\035ow)g(e)-5 b(quivalent)40 b Fn(i\033)e(there)g(exists)f(a)h(homeomorphism)i Fl(h)e Fi(:)g Fl(M)48 b Fk(!)38 b Fl(N)581 1000 y Fn(suc)m(h)31 b(that)1763 1112 y Fl(\036)1817 1126 y Fj(t)1872 1112 y Fi(=)25 b Fl(h)2020 1075 y Ff(\000)p Fg(1)2135 1112 y Fk(\001)c Fl( )2240 1126 y Fj(t)2290 1112 y Fk(\001)f Fl(h)581 1274 y Fn(for)30 b(all)h Fl(t)25 b Fk(2)g Fn(I)-14 b(R)n(.)722 1499 y Fm(Remark:)61 b Fn(This)41 b(ob)m(viously)g(corresp) s(onds)g(to)g(top)s(ological)h(conjugacy)f(in)g(the)581 1612 y(time)30 b(-)e(discrete)i(case,)f(since)g(the)h(de\034ning)g (equation)f(for)g(the)g(conjugacy)h(also)f(holds)581 1725 y(for)39 b(iterates)h(of)f Fl(f)5 b(;)15 b(g)s Fn(.)67 b(But)39 b(\035o)m(w)h(equiv)-5 b(alence)40 b(is,)h(in)e(general,)k(to) s(o)c(restrictiv)m(e.)68 b(If)581 1838 y(one)41 b(is)g(in)m(terested)h (in)f(the)g(dynamical)g(b)s(eha)m(viour)h(of)f(\035o)m(ws,)j(lik)m(e)d (the)g(n)m(um)m(b)s(er)h(of)581 1951 y(equilibria,)32 b(one)e(should)h(at)g(least)f(allo)m(w)i(time)e(shifts.)722 2177 y Fm(Def:)39 b Fn(Let)30 b Fl(\036)1164 2191 y Fj(t)1194 2177 y Fl(;)15 b( )1293 2191 y Fj(t)1351 2177 y Fn(b)s(e)29 b(\035o)m(ws)g(on)h Fl(M)10 b Fn(.)39 b Fl( )2041 2191 y Fj(t)2100 2177 y Fn(is)28 b(called)i(a)f Fh(time)i(change)f Fn(of)e Fl(\036)3173 2191 y Fj(t)3231 2177 y Fn(i\033)h(for)g(all)581 2290 y Fl(x)c Fk(2)g Fl(M)40 b Fn(the)31 b(orbits)g(coincide,)g(i.e.,) 1385 2483 y Fk(f)p Fl(\036)1484 2497 y Fj(t)1515 2483 y Fi(\()p Fl(x)p Fi(\))26 b(:)f Fl(t)g Fk(2)g Fn(I)-14 b(R)o Fk(g)51 b Fi(=)f Fk(f)p Fl( )2263 2497 y Fj(t)2293 2483 y Fi(\()p Fl(x)p Fi(\))26 b(:)g Fl(t)f Fk(2)g Fn(I)-14 b(R)n Fk(g)581 2675 y Fn(and)31 b(the)g(\035o)m(w)g(orien)m(tation)i (is)c(preserv)m(ed.)722 2901 y Fm(Def:)54 b Fl(M)5 b(;)15 b(N)48 b Fn(as)37 b(b)s(efore.)62 b(T)-8 b(w)m(o)39 b Fl(C)1939 2868 y Fj(r)2014 2901 y Fn(-)e(\035o)m(ws)h Fl(\036)2366 2915 y Fj(t)2433 2901 y Fi(:)g Fl(M)47 b Fk(!)37 b Fl(M)5 b(;)15 b( )2951 2915 y Fj(t)3018 2901 y Fi(:)38 b Fl(N)47 b Fk(!)37 b Fl(N)47 b Fn(are)581 3014 y(called)37 b Fl(C)915 2981 y Fg(0)992 3014 y Fh(-)h(orbit)g(e)-5 b(quivalent)39 b Fn(i\033)d(there)h(exists)e(a)i(homeomorphism)h Fl(h)e Fi(:)f Fl(M)46 b Fk(!)35 b Fl(N)581 3127 y Fn(suc)m(h)c(that) 1768 3240 y Fl(\027)1813 3254 y Fj(t)1868 3240 y Fi(=)25 b Fl(h)2016 3202 y Ff(\000)p Fg(1)2131 3240 y Fk(\001)20 b Fl( )2235 3254 y Fj(t)2285 3240 y Fk(\001)h Fl(h)581 3401 y Fn(is)30 b(a)g(time)h(c)m(hange)h(of)e Fl(\036)1409 3415 y Fj(t)1439 3401 y Fn(.)40 b(This)31 b(is)f(equiv)-5 b(alen)m(t)31 b(to)f(requiring)i(that)1324 3594 y Fl(h)p Fi(\()p Fk(f)p Fl(\036)1510 3608 y Fj(t)1541 3594 y Fi(\()p Fl(x)p Fi(\))26 b(:)f Fl(t)g Fk(2)g Fn(I)-14 b(R)o Fk(g)p Fi(\))51 b(=)f Fk(f)p Fl( )2324 3608 y Fj(t)2355 3594 y Fi(\()p Fl(x)p Fi(\))26 b(:)f Fl(t)g Fk(2)g Fn(I)-14 b(R)o Fk(g)581 3786 y Fn(and)31 b Fl(h)f Fn(preserv)m(es)h(the)g(\035o) m(w)g(orien)m(tation.)722 4012 y Fm(Def:)40 b Fn(A)29 b Fl(C)1119 3979 y Fj(r)1187 4012 y Fn(-)h(\035o)m(w)h Fl(\036)1489 4026 y Fj(t)1548 4012 y Fn(on)f Fl(M)40 b Fn(is)30 b(called)g Fl(C)2219 3979 y Fj(r)2289 4012 y Fh(-)i(\(str)-5 b(ongly\))32 b(structur)-5 b(al)5 b(ly)33 b(stable)e Fn(i\033)581 4125 y(there)36 b(exists)f(a)h(neigh)m(b)s (orho)s(o)s(d)g Fl(U)46 b Fn(of)35 b Fl(\036)1995 4139 y Fj(t)2060 4125 y Fn(\(w.r.t.)57 b(the)35 b Fi(\000)2579 4092 y Fj(r)2617 4125 y Fi(\()p Fl(M)10 b Fi(\))36 b Fn(-)g(top)s(ology)g(induced)581 4238 y(b)m(y)e(the)g(v)m(ector)g (\034elds\))g(suc)m(h)g(that)f(eac)m(h)i(\035o)m(w)f Fl( )2275 4252 y Fj(t)2335 4238 y Fk(2)c Fl(U)44 b Fn(is)33 b Fl(C)2698 4205 y Fg(0)2770 4238 y Fn(-)g(orbit)h(equiv)-5 b(alen)m(t)35 b(to)581 4351 y Fl(\036)635 4365 y Fj(t)695 4351 y Fn(\()p Fi(\()p Fl( )824 4365 y Fj(n)872 4351 y Fi(\))907 4365 y Fj(t)962 4351 y Fk(!)25 b Fl(\036)1132 4365 y Fj(t)1192 4351 y Fn(in)31 b Fi(\000)1355 4318 y Fj(r)1392 4351 y Fi(\()p Fl(M)10 b Fi(\))31 b Fk(\))f Fl(h)1764 4365 y Fj(n)1837 4351 y Fk(!)25 b Fi(id)k Fn(in)i(the)f Fl(C)2392 4318 y Fg(0)2461 4351 y Fn(-)h(top)s(ology)-8 b(,)31 b(resp\).)581 4749 y Fo(T)-11 b(o)l(y)45 b(mo)t(del:)57 b(Flo)l(w)44 b(on)g Fe(D)1883 4705 y Fd(n)581 4951 y Fn(In)30 b(this)g(section)h(w)m(e)g(consider)g(\035o)m(ws)g(on)g(the)g (unit)g(ball)581 5064 y Fl(D)659 5031 y Fj(n)756 5064 y Fi(:=)51 b Fk(f)p Fl(x)g Fk(2)f Fn(I)-14 b(R)1247 5028 y Fj(n)1344 5064 y Fi(:)51 b Fk(j)p Fl(x)p Fk(j)g(\024)f Fi(1)p Fk(g)h(\032)f Fn(I)-14 b(R)2041 5028 y Fj(n)2088 5064 y Fn(.)85 b(In)46 b(order)g(to)f(ensure)h(that)g(the)g(\035o)m(w-) 581 5177 y(lines)37 b(sta)m(y)h(in)f Fl(D)1183 5144 y Fj(n)1267 5177 y Fn(for)g(all)g(p)s(ositiv)m(e)h(times,)h(w)m(e)f(imp)s (ose)f(the)h(natural)g(requiremen)m(t)581 5290 y Fl(<)25 b(X)7 b Fi(\()p Fl(x)p Fi(\))p Fl(;)15 b(x)27 b(>)e(<)50 b Fi(0)31 b Fn(for)f(all)g Fl(x)c Fk(2)e Fl(@)5 b(D)1850 5257 y Fj(n)1928 5290 y Fn(on)30 b(all)h(v)m(ector)g(\034elds)g Fl(X)h Fi(:)26 b Fl(D)2918 5257 y Fj(n)2990 5290 y Fk(!)f Fn(I)-14 b(R)3191 5254 y Fj(n)3238 5290 y Fn(.)p Black Black eop %%Page: 3 3 3 2 bop Black 3126 249 a Fn(3)p Black 324 548 a(The)37 b(con)m(ten)m(t)h(of)e(this)h(section)g(en)m(tirely)g(consists)f(of)g (stating)h(and)g(pro)m(ving)h(the)183 661 y(follo)m(wing)31 b(theorem:)324 905 y Fm(Theorem:)40 b Fn(Let)31 b Fl(\036)1015 919 y Fj(t)1075 905 y Fn(b)s(e)f(a)g Fl(C)1345 873 y Fg(1)1414 905 y Fn(-)g(\035o)m(w)i(on)e Fl(D)1866 873 y Fj(n)1943 905 y Fn(suc)m(h)h(that)183 1018 y(-)f Fl(\036)297 1032 y Fj(t)357 1018 y Fn(has)g(exactly)g(one)h(equilibrium)h Fi(0)26 b Fk(2)e Fl(D)1709 985 y Fj(n)1756 1018 y Fn(,)30 b(whic)m(h)i(is)e(a)g(sink,)183 1131 y(-)g Fi(lim)369 1145 y Fj(t)p Ff(!1)555 1131 y Fl(\036)609 1145 y Fj(t)639 1131 y Fi(\()p Fl(x)p Fi(\))c(=)f(0)30 b Fn(for)h(all)f Fl(x)25 b Fk(2)g Fl(D)1463 1098 y Fj(n)1510 1131 y Fn(.)183 1244 y(Then)31 b Fl(\036)473 1258 y Fj(t)533 1244 y Fn(is)f Fl(C)696 1211 y Fg(1)765 1244 y Fn(-)g(structurally)h(stable.)324 1489 y(The)g(pro)s(of)f(consists)g(of)g(three)h(steps:)324 1733 y Fh(First)k(step:)46 b Fn(There)35 b(exists)e(a)g(neigh)m(b)s (orho)s(o)s(d)i Fl(U)41 b Fk(\032)31 b Fi(\000)2208 1701 y Fg(1)2247 1733 y Fi(\()p Fl(M)10 b Fi(\))34 b Fn(of)f Fl(\036)2609 1747 y Fj(t)2673 1733 y Fn(and)h(a)f(neigh-)183 1846 y(b)s(orho)s(o)s(d)f Fl(V)52 b Fn(of)33 b Fi(0)c Fk(2)g Fl(D)997 1813 y Fj(n)1076 1846 y Fn(suc)m(h)k(that)g(eac)m(h)g Fl( )1744 1860 y Fj(t)1803 1846 y Fk(2)c Fl(U)42 b Fn(has)32 b(a)h(unique)g(equilibrium)h Fl(a)e Fn(in)183 1959 y Fl(V)20 b Fn(,)30 b(whic)m(h)i(is)e(again)h(a)f(sink.)324 2204 y(This)23 b(observ)-5 b(ation)25 b(relies)f(on)f(the)h(follo)m (wing)h(generalized)h(v)m(ersion)e(of)f(the)h(implicit)183 2317 y(function)31 b(theorem:)324 2562 y Fm(Lemma:)48 b Fn(Let)35 b Fl(W)44 b Fk(\032)31 b Fn(I)-14 b(R)1217 2525 y Fj(n)1298 2562 y Fn(b)s(e)34 b(op)s(en,)h Fl(X)k Fi(:)32 b Fl(W)45 b Fk(!)31 b Fn(I)-14 b(R)2182 2525 y Fj(n)2263 2562 y Fn(a)34 b Fl(C)2414 2529 y Fg(1)2487 2562 y Fn(-)g(v)m(ector)h(\034eld)g(and)183 2674 y Fl(x)235 2688 y Fg(0)306 2674 y Fk(2)c Fl(W)13 b Fn(.)51 b(Assume)33 b(that)i Fl(D)s(X)7 b Fi(\()p Fl(x)1358 2688 y Fg(0)1398 2674 y Fi(\))32 b(:)g Fn(I)-14 b(R)1606 2638 y Fj(n)1685 2674 y Fk(!)31 b Fn(I)-14 b(R)1892 2638 y Fj(n)1973 2674 y Fn(is)33 b(in)m(v)m(ertible.)54 b(Then)35 b(there)g(exist)183 2787 y(op)s(en)d(neigh)m(b)s(orho)s(o)s(ds)h Fl(U)39 b Fk(\032)28 b Fi(\000)1265 2754 y Fg(1)1305 2787 y Fi(\()p Fl(M)10 b Fi(\))33 b Fn(of)f Fl(X)39 b Fn(and)33 b Fl(V)52 b Fn(of)32 b Fl(x)2165 2801 y Fg(0)2237 2787 y Fn(suc)m(h)h(that)g(for) f(all)h Fl(Y)48 b Fk(2)28 b Fl(U)183 2900 y Fn(w)m(e)j(ha)m(v)m(e)183 3013 y(-)f Fl(Y)20 b Fk(j)p Fl(V)51 b Fn(is)29 b(injectiv)m(e,)183 3126 y(-)h Fl(X)7 b Fi(\()p Fl(x)412 3140 y Fg(0)452 3126 y Fi(\))26 b Fk(2)f Fl(Y)20 b Fi(\()p Fl(V)g Fi(\))p Fn(.)324 3371 y Fm(Remark:)37 b Fn(The)27 b(ab)s(o)m(v)m(e)h(statemen)m (t)g(follo)m(ws)f(from)g(the)f(lemma)i(since)f Fl(D)s(X)7 b Fi(\(0\))27 b Fn(for)183 3484 y(a)g(sink)g(is)g(negativ)m(e)i (de\034nite)g(and)f(th)m(us)g(clearly)g(in)m(v)m(ertible.)42 b(It)27 b(follo)m(ws)h(that)g(one)g(can)183 3597 y(c)m(ho)s(ose)33 b(a)g(unique)h Fl(a)c Fk(2)f Fl(V)53 b Fn(suc)m(h)34 b(that)g Fi(0)c(=)g Fl(X)7 b Fi(\(0\))31 b(=)f Fl(Y)20 b Fi(\()p Fl(a)p Fi(\))p Fn(.)49 b(Here)33 b Fl(U)43 b Fn(can)34 b(b)s(e)e(c)m(hosen)183 3709 y(so)f(small)i(suc)m(h)f(that) h Fk(j)p Fl(a)21 b Fk(\000)g Fi(0)p Fk(j)29 b Fl(<)f(\017)k Fn(\(for)g(arbitrary)h(small)f Fl(\017)c(>)g Fi(0)p Fn(\))k(for)g(all)g Fl(Y)48 b Fk(2)28 b Fl(U)42 b Fn(and)183 3822 y(suc)m(h)31 b(that)f Fl(a)h Fn(has)f(the)h(same)f(index)h(for)f Fl(Y)50 b Fn(as)30 b(0)g(has)g(for)h Fl(X)7 b Fn(.)324 4067 y Fh(Se)-5 b(c)g(ond)39 b(step:)51 b Fn(All)36 b(the)g(p)s(ositiv)m(e)g (\035o)m(wlines)h(of)f Fl( )2055 4081 y Fj(t)2119 4067 y Fk(2)f Fl(U)45 b Fn(end)37 b(in)f(the)g(asso)s(ciated)183 4180 y(sink)29 b Fl(a)i Fn(\(with)g Fl(U)40 b Fn(p)s(ossibly)29 b(smaller\).)324 4425 y Fm(Lemma:)60 b Fn(There)40 b(exists)g(a)f(n)m (um)m(b)s(er)j Fl(r)i(>)d Fi(0)f Fn(and)h(a)f(p)s(ossibly)f(smaller)h (neigh-)183 4537 y(b)s(orho)s(o)s(d)d Fl(U)47 b Fn(of)37 b Fl(X)44 b Fn(\()37 b(=)h(\035o)m(w)g(of)f Fl(\036)1426 4551 y Fj(t)1493 4537 y Fn(\))g(suc)m(h)h(that)g(for)f(eac)m(h)h Fl(Y)57 b Fk(2)36 b Fl(U)47 b Fn(w)m(e)38 b(ha)m(v)m(e)h(that)183 4650 y Fl(a)25 b Fk(2)g Fl(B)411 4664 y Fj(r)474 4650 y Fi(:=)g Fk(f)p Fl(x)h Fk(2)f Fl(D)882 4617 y Fj(n)954 4650 y Fi(:)g Fk(j)p Fl(x)p Fk(j)h(\024)f Fl(r)s Fk(g)p Fn(,)i Fl(B)1438 4664 y Fj(r)1502 4650 y Fn(is)f(in)g(the)h(basin)f(of) g Fl(a)g Fn(and)h(p)s(ositiv)m(ely)f(in)m(v)-5 b(arian)m(t)183 4763 y(under)31 b(the)f(\035o)m(w)i Fl( )842 4777 y Fj(t)902 4763 y Fn(of)e Fl(Y)20 b Fn(.)183 4876 y(\()p Fh(b)-5 b(asin)30 b Fn(of)g Fl(a)g Fn(:=)h Fk(f)p Fl(x)26 b Fk(2)e Fl(D)1039 4843 y Fj(n)1111 4876 y Fi(:)i(lim)1288 4890 y Fj(t)p Ff(!1)1474 4876 y Fl( )1533 4890 y Fj(t)1563 4876 y Fi(\()p Fl(x)p Fi(\))g(=)f Fl(a)p Fk(g)p Fn(\))324 5121 y Fh(Pr)-5 b(o)g(of:)58 b Fn(De\034ne)39 b Fi(2)p Fl(\027)46 b Fi(:=)39 b(max)p Fk(f)p Fl(<)h(D)s(X)7 b Fi(\(0\))27 b Fk(\001)g Fl(v)s(;)15 b(v)43 b(>)p Fi(:)d Fk(j)p Fl(v)s Fk(j)g Fi(=)f(1)p Fk(g)i Fl(<)e Fi(0)p Fn(.)66 b(Hence)40 b(w)m(e)183 5234 y(obtain)1088 5403 y Fl(<)25 b(D)s(X)7 b Fi(\(0\))21 b Fk(\001)g Fl(x;)15 b(x)25 b(>)50 b Fk(\024)76 b Fi(2)p Fl(\027)26 b Fk(\001)21 b(j)p Fl(x)p Fk(j)2227 5366 y Fg(2)p Black Black eop %%Page: 4 4 4 3 bop Black 581 249 a Fn(4)p Black 581 548 a(so)30 b(that)h(w)m(e)g(can)g(\034nd)g(some)f Fl(r)e(>)d Fi(0)30 b Fn(suc)m(h)h(that)1603 755 y Fl(<)25 b(X)7 b Fi(\()p Fl(x)p Fi(\))p Fl(;)15 b(x)27 b(>)50 b Fk(\024)76 b Fl(\027)26 b Fk(\001)20 b(j)p Fl(x)p Fk(j)2508 718 y Fg(2)581 962 y Fn(for)30 b(all)h Fl(x)25 b Fk(2)g Fl(B)1077 976 y Fg(2)p Fj(r)1150 962 y Fn(.)40 b(No)m(w)31 b(for)f Fl(U)36 b Fk(\032)25 b Fi(\000)1810 929 y Fg(1)1849 962 y Fi(\()p Fl(M)10 b Fi(\))31 b Fn(small)f(enough,)i(one)f(can)f(pro)m(v)m(e)i (that)1278 1170 y Fl(<)25 b(Y)20 b Fi(\()p Fl(x)p Fi(\))p Fl(;)15 b(x)21 b Fk(\000)f Fl(a)25 b(>)51 b Fk(\024)75 b Fl(\026)20 b Fk(\001)g(j)p Fl(x)h Fk(\000)f Fl(a)p Fk(j)2496 1132 y Fg(2)2561 1170 y Fl(;)66 b(\026)25 b(<)g Fi(0)581 1377 y Fn(for)j(all)g Fl(x)d Fk(2)g Fl(B)1072 1391 y Fj(r)1137 1377 y Fn(and)j Fl(Y)46 b Fk(2)24 b Fl(U)10 b Fn(.)40 b(This)28 b(is)f(the)h(p)s(oin)m(t)g(where)h(one)f (can)g(mak)m(e)h(explicit)f(use)581 1490 y(of)i(the)h(prop)s(ert)m(y)g (of)f Fl(Y)50 b Fn(of)30 b(b)s(eing)h Fl(C)1834 1457 y Fg(1)1903 1490 y Fn(-)f(close.)41 b(W)-8 b(e)30 b(start)h(with)611 1697 y Fl(<)25 b(Y)20 b Fi(\()p Fl(x)p Fi(\))p Fl(;)15 b(x)21 b Fk(\000)f Fl(a)25 b(>)50 b Fi(=)g Fl(<)25 b(X)7 b Fi(\()p Fl(x)21 b Fk(\000)f Fl(a)p Fi(\))p Fl(;)15 b(x)21 b Fk(\000)f Fl(a)25 b(>)51 b Fi(+)f Fl(<)25 b(Y)20 b Fi(\()p Fl(x)p Fi(\))h Fk(\000)f Fl(X)7 b Fi(\()p Fl(x)20 b Fk(\000)g Fl(a)p Fi(\))p Fl(;)15 b(x)21 b Fk(\000)f Fl(a)26 b(>)581 1904 y Fn(If)j Fl(U)40 b Fn(is)30 b(small)g(enough)h (suc)m(h)g(that)f Fl(a)25 b Fk(2)g Fl(B)2038 1918 y Fj(\017)2071 1904 y Fi(\(0\))31 b Fn(with)g Fl(\017)25 b(<)g(r)32 b Fn(the)e(\034rst)h(addend)g(can)f(b)s(e)581 2017 y(b)s(ounded)j(b)m (y)g Fl(\027)6 b Fk(j)p Fl(x)22 b Fk(\000)f Fl(a)p Fk(j)1395 1984 y Fg(2)1467 2017 y Fn(for)32 b(all)h Fl(x)c Fk(2)f Fl(B)1974 2031 y Fj(r)2012 2017 y Fn(.)47 b(Hence)33 b(it)f(su\036ces)h(to)g(b)s(ound)f(the)h(second)581 2130 y(appropriately)-8 b(.)43 b(Using)30 b(Cauc)m(h)m(y-Sc)m(h)m(w)m(arz)35 b(w)m(e)c(ha)m(v)m(e)895 2337 y Fl(<)25 b(Y)20 b Fi(\()p Fl(x)p Fi(\))h Fk(\000)f Fl(X)7 b Fi(\()p Fl(x)21 b Fk(\000)f Fl(a)p Fi(\))p Fl(;)15 b(x)21 b Fk(\000)f Fl(a)25 b(>)50 b Fk(\024)76 b(j)p Fl(Y)20 b Fi(\()p Fl(x)p Fi(\))h Fk(\000)f Fl(X)7 b Fi(\()p Fl(x)21 b Fk(\000)f Fl(a)p Fi(\))p Fk(j)h(\001)f(j)p Fl(x)g Fk(\000)g Fl(a)p Fk(j)581 2545 y Fn(so)25 b(that)g(w)m(e)g(just) f(need)i(to)f(mak)m(e)g(use)f(of)h(the)g(Lipsc)m(hitz)h(prop)s(ert)m(y) g(of)e Fl(Z)7 b Fi(\()p Fl(x)p Fi(\))26 b(:=)f Fl(Y)20 b Fi(\()p Fl(x)p Fi(\))9 b Fk(\000)581 2658 y Fl(X)e Fi(\()p Fl(x)21 b Fk(\000)f Fl(a)p Fi(\))31 b Fn(\()p Fl(Z)36 b Fn(is)30 b Fl(C)1272 2625 y Fg(1)1311 2658 y Fn(\),)1603 2773 y Fk(j)p Fl(Z)7 b Fi(\()p Fl(x)p Fi(\))20 b Fk(\000)g Fi(0)p Fk(j)52 b(\024)e Fl(\021)23 b Fk(\001)e(j)p Fl(x)f Fk(\000)g Fl(a)p Fk(j)581 2944 y Fn(By)34 b(c)m(ho)s(osing)h Fl(V)55 b Fn(small)34 b(enough,)j(w)m(e)e(obtain)g Fl(\021)h(<)31 b Fk(\000)p Fl(\027)40 b Fn(and)35 b(th)m(us)g Fl(\026)c Fi(:=)h Fl(\027)d Fi(+)23 b Fl(\021)37 b Fn(do)s(es)581 3057 y(the)31 b(job.)722 3284 y(All)g(the)h(statemen)m(ts)h(in)e(the)h (theorem)h(no)m(w)f(follo)m(w)g(from)g(the)g(ab)s(o)m(v)m(e)g (inequalit)m(y)581 3396 y(for)e Fl(Y)20 b Fn(.)41 b(q.e.d.)722 3623 y(So)33 b(far)g(w)m(e)g(ha)m(v)m(e)h(sho)m(wn)g(that)f(some)g (small)g(neigh)m(b)s(orho)s(o)s(d)h Fl(B)2919 3637 y Fj(r)2989 3623 y Fn(of)f(the)g(sink)f(lies)581 3736 y(in)38 b(the)g(basin)h(of)e Fl(a)h Fn(for)g(all)g Fl(Y)58 b Fk(2)37 b Fl(U)10 b Fn(.)63 b(It)38 b(remains)g(to)g(pro)m(v)m(e)h (that)g(this)f(extends)g(to)581 3849 y(all)31 b(of)f Fl(D)888 3816 y Fj(n)935 3849 y Fn(.)41 b(But)30 b(this)h(can)g(sho)m (wn)g(b)m(y)g(simple)g(con)m(tin)m(uit)m(y)i(argumen)m(ts,)f(since)f (all)f(the)581 3962 y(equilibria)h(lie)g(in)g Fl(B)1276 3976 y Fj(r)1313 3962 y Fn(.)722 4189 y(F)-8 b(or)30 b(all)f Fl(x)c Fk(2)p 1170 4116 343 4 v 25 w Fl(D)1248 4163 y Fj(n)1315 4189 y Fk(\000)20 b Fl(B)1475 4203 y Fj(r)1542 4189 y Fn(there)30 b(exists)e(some)h Fl(t)c Fk(2)g Fn(I)-14 b(R)28 b(suc)m(h)h(that)h Fl(\036)2954 4203 y Fj(t)2984 4189 y Fi(\()p Fl(x)p Fi(\))c Fk(2)e Fl(B)3286 4203 y Fj(r)3324 4189 y Fn(,)30 b(since)581 4302 y Fl(D)659 4269 y Fj(n)741 4302 y Fn(is)35 b(in)g(the)g(basin)h (of)e Fl(\036)1510 4316 y Fj(t)1540 4302 y Fn(.)55 b(Since)36 b(this)f(set)g(is)g(compact,)i(w)m(e)f(\034nd)g(some)f(maximal)581 4415 y(time)j Fl(t)827 4429 y Fg(0)866 4415 y Fn(.)61 b(It)37 b(follo)m(ws)h(from)g(con)m(tin)m(uit)m(y)h(that)f Fl( )2277 4429 y Fj(t)2307 4415 y Fi(\()p 2342 4342 V Fl(D)2420 4388 y Fj(n)2487 4415 y Fk(\000)20 b Fl(B)2647 4429 y Fj(r)2685 4415 y Fi(\))38 b Fk(\032)e Fl(B)2934 4429 y Fj(r)3009 4415 y Fn(for)h Fl(t)g Fk(\025)g Fl(t)3365 4429 y Fg(0)3429 4415 y Fi(+)24 b(1)581 4528 y Fn(for)30 b Fl(Y)49 b Fn(in)30 b(some)g(\(once)h(again\))g(p)s(ossibly)e(smaller) h Fl(U)35 b Fk(\032)25 b Fi(\000)2577 4495 y Fg(1)2616 4528 y Fi(\()p Fl(M)10 b Fi(\))p Fn(.)41 b(Therefore)31 b(w)m(e)g(ha)m(v)m(e)581 4641 y(sho)m(wn)g(for)f(all)h Fl(Y)45 b Fk(2)25 b Fl(U)40 b Fn(that)1814 4756 y Fi(lim)1794 4812 y Fj(t)p Ff(!1)1976 4756 y Fl( )2035 4770 y Fj(t)2065 4756 y Fi(\()p Fl(x)p Fi(\))26 b(=)f Fl(a)581 4950 y Fn(for)30 b(all)h Fl(x)25 b Fk(2)g Fl(D)1086 4917 y Fj(n)1163 4950 y Fn(with)31 b Fl( )1428 4964 y Fj(t)1488 4950 y Fn(b)s(eing)f(the)h(\035o)m(w)g(of)f Fl(Y)20 b Fn(.)722 5177 y Fh(Thir)-5 b(d)33 b(step:)41 b Fn(Construction)32 b(of)e(the)h(conjugacy)g(homeomorphism)i Fl(h)26 b Fk(2)g Fl(C)3336 5144 y Fg(0)3374 5177 y Fi(\()p Fl(D)3487 5144 y Fj(n)3535 5177 y Fi(\))581 5290 y Fn(for)k(the)h(orbits)g(of)f Fl(\036)1289 5304 y Fj(t)1349 5290 y Fn(and)g Fl( )1583 5304 y Fj(t)1613 5290 y Fn(.)40 b(This)31 b(pro)m(v)m(es)g(the)g (structural)g(stabilit)m(y)-8 b(.)p Black Black eop %%Page: 5 5 5 4 bop Black 3126 249 a Fn(5)p Black 324 548 a(It)35 b(needs)h(to)g(b)s(e)g(sho)m(wn)h(that)f Fl(\036)1459 562 y Fj(t)1525 548 y Fn(is)f Fl(C)1693 515 y Fg(0)1768 548 y Fn(-)g(orbit)i(equiv)-5 b(alen)m(t)37 b(to)f(all)g(\035o)m(ws)h Fl( )3031 562 y Fj(t)3096 548 y Fn(in)183 661 y(the)c(neigh)m(b)s(orho) s(o)s(d)h Fl(U)10 b Fn(.)49 b(The)34 b(required)g(homeomorphism)h(can)f (b)s(e)f(determined)h(ex-)183 774 y(plicitely)26 b(b)m(y)g(means)h(of)e (the)i(\035o)m(ws)f Fl(\036)1439 788 y Fj(t)1495 774 y Fn(and)g Fl( )1725 788 y Fj(t)1755 774 y Fn(.)39 b(W)-8 b(e)26 b(will)g(not)h(only)f(end)g(up)g(with)h(orbit)183 887 y(equiv)-5 b(alence)31 b(but)g(ev)m(en)f(with)h(\035o)m(w)h(equiv) -5 b(alence:)324 1118 y(De\034ne)39 b(maps)f Fi(\010)g(:)h([0)p Fl(;)15 b Fk(1)p Fi(\))26 b Fk(\002)g Fl(@)5 b(D)1514 1085 y Fj(n)1599 1118 y Fk(!)38 b Fl(D)1806 1085 y Fj(n)1878 1118 y Fk(\000)26 b(f)p Fi(0)p Fk(g)39 b Fn(and)f Fi(\011)h(:)f([0)p Fl(;)15 b Fk(1)p Fi(\))27 b Fk(\002)e Fl(@)5 b(D)2995 1085 y Fj(n)3081 1118 y Fk(!)183 1231 y Fl(D)261 1198 y Fj(n)330 1231 y Fk(\000)22 b(f)p Fl(a)p Fk(g)35 b Fn(b)m(y)f Fi(\010\()p Fl(t;)15 b(x)p Fi(\))32 b(=)e Fl(\036)1173 1245 y Fj(t)1203 1231 y Fi(\()p Fl(x)p Fi(\))35 b Fn(and)f Fi(\011\()p Fl(t;)15 b(x)p Fi(\))32 b(=)e Fl( )1997 1245 y Fj(t)2027 1231 y Fi(\()p Fl(x)p Fi(\))p Fn(.)52 b(These)34 b(maps)g(are)g(homeo-)183 1343 y(morphisms.)45 b(Note)32 b(esp)s(ecially)g(that)g(for)f(ev)m(ery)h(p)s(oin)m(t)h Fl(D)2201 1310 y Fj(n)2269 1343 y Fk(\000)21 b(f)p Fi(sink)o Fk(g)32 b Fn(there)g(exists)f(a)183 1456 y(unique)f(p)s(oin)m(t)h(on)g (the)g(b)s(oundary)-8 b(,)31 b(whic)m(h)h('\035o)m(ws')f(to)f(it)h(in)f (a)h(\(unique\))g(\034nite)g(time.)324 1687 y(With)d(these)g(w)m(e)h (can)g(construct)g(the)f(conjugacy)h Fl(h)d Fi(:)f Fl(D)2253 1654 y Fj(n)2325 1687 y Fk(!)g Fl(D)2519 1654 y Fj(n)2594 1687 y Fn(easily)-8 b(.)40 b(Namely)183 1800 y(set)30 b Fl(h)p Fi(\()p Fl(y)s Fi(\))c(=)f(\(\011)20 b Fk(\001)h Fi(\010)854 1767 y Ff(\000)p Fg(1)948 1800 y Fi(\)\()p Fl(y)s Fi(\))31 b Fn(for)f Fl(y)e Fk(6)p Fi(=)d(0)30 b Fn(and)h Fl(h)p Fi(\(0\))c(=)e Fl(a)p Fn(.)183 2116 y Fo(Structural)45 b(stabilit)l(y)h(of)e(Anoso)l(v)g(di\033eomorphisms) 183 2329 y Fn(W)-8 b(ell,)31 b(what)g(migh)m(t)g(the)g(con)m(ten)m(t)h (of)e(this)h(last)f(section)h(?)40 b(Y)-8 b(es,)30 b(y)m(ou)h(are)f (righ)m(t.)324 2560 y Fm(Theorem:)39 b Fn(Anoso)m(v)30 b(di\033eos)g(on)g(compact)h(manifolds)f(are)g(\()p Fl(C)2548 2527 y Fg(1)2617 2560 y Fn(-\))f(structurally)183 2673 y(stable.)324 2904 y Fm(Pro)s(of:)63 b Fn(W)-8 b(e)43 b(pro)m(v)m(e)g(this)g(b)m(y)f(means)h(of)f(the)h(implicit)h(function)f (theorem)g(for)183 3017 y(Banac)m(h)d(spaces.)68 b(Since)40 b(it)f(is)g(a)h(purely)f(lo)s(cal)h(statemen)m(t,)j(it)d(applies)f(to)h (Banac)m(h)183 3129 y(manifolds)34 b(as)f(w)m(ell.)52 b(Ho)m(w)m(ev)m(er,)37 b(w)m(e)d(use)g(here)g(a)g(v)m(ersion)g(with)h (relaxed)f(conditions.)183 3242 y(T)-8 b(o)31 b(b)s(e)f(more)h (precise,)g(our)f(pro)s(of)g(relies)h(on)g(the)f(follo)m(wing)i(lemma:) 324 3473 y Fm(Lemma:)50 b Fn(Let)36 b Fl(E)41 b Fn(b)s(e)35 b(a)g(top)s(ological)i(v)m(ector)f(space,)h Fl(F)49 b Fn(a)35 b(Banac)m(h)i(space.)56 b(Let)183 3586 y Fl(f)34 b Fi(:)26 b Fl(U)15 b Fk(\002)5 b Fl(V)45 b Fk(!)25 b Fl(F)35 b Fn(\()p Fl(U)h Fk(\032)25 b Fl(E)5 b(;)15 b(V)46 b Fk(\032)25 b Fl(F)13 b Fn(\))23 b(b)s(e)f Fl(C)1625 3553 y Fg(1)1686 3586 y Fn(with)i(in)m(v)m(ertible)g(F)-8 b(rec)m(het)25 b(deriv)-5 b(ativ)m(e)24 b(with)183 3699 y(resp)s(ect)31 b(to)h(the)f(second)h(co)s(ordinate.)45 b(Then)32 b(there)h(exists)d(a)i(mapping)g Fl(g)f Fi(:)c Fl(U)2894 3713 y Fg(1)2961 3699 y Fk(!)g Fl(V)3132 3713 y Fg(1)183 3812 y Fn(\()p Fl(x)270 3826 y Fg(0)338 3812 y Fk(2)h Fl(U)489 3826 y Fg(1)558 3812 y Fk(\032)g Fl(U;)15 b(y)804 3826 y Fg(0)872 3812 y Fk(2)29 b Fl(V)1015 3826 y Fg(1)1083 3812 y Fk(\032)g Fl(V)20 b Fn(\))32 b(suc)m(h)h(that)g Fl(f)10 b Fi(\()p Fl(x;)15 b(y)s Fi(\))29 b(=)g Fl(f)10 b Fi(\()p Fl(x)2263 3826 y Fg(0)2302 3812 y Fl(;)15 b(y)2387 3826 y Fg(0)2426 3812 y Fi(\))29 b Fk(\()-15 b(\))29 b Fl(y)i Fi(=)e Fl(g)s Fi(\()p Fl(x)p Fi(\))k Fn(for)183 3925 y(all)d Fl(x)25 b Fk(2)g Fl(U)533 3939 y Fg(1)573 3925 y Fl(;)15 b(y)28 b Fk(2)d Fl(V)825 3939 y Fg(1)865 3925 y Fn(.)324 4156 y Fm(Pro)s(of)38 b(of)h(the)h(lemma:)48 b Fn(see)34 b(the)h(pap)s(er)f Fh(H.)i(Glo)-5 b(e)g(ckner,)39 b(Implicit)d(F)-7 b(unctions)183 4269 y(fr)i(om)32 b(T)-7 b(op)i(olo)g(gic)g(al)34 b(V)-7 b(e)i(ctor)34 b(Sp)-5 b(ac)g(es)34 b(to)e(Banach)h(Sp)-5 b(ac)g(es,)34 b (arxiv:math.GM/0303320)p Fn(.)324 4500 y(Therefore,)46 b(let)c Fl(D)47 b Fi(:=)e(Di\013)7 b(\()p Fl(M)j Fi(\))p Fn(,)45 b Fl(C)51 b Fi(:=)45 b Fl(C)1900 4467 y Fg(0)1939 4500 y Fi(\()p Fl(M)5 b(;)15 b(M)10 b Fi(\))43 b Fn(and)f(de\034ne)h(a) f(mapping)183 4613 y Fl(A)25 b Fi(:)h Fl(D)21 b Fk(\002)d Fl(C)24 b Fk(\002)18 b Fl(D)28 b Fk(!)d Fl(C)g Fk(\002)18 b Fl(C)36 b Fn(with)30 b Fl(A)p Fi(\()p Fl(g)1540 4627 y Fg(1)1580 4613 y Fl(;)15 b(h;)g(g)1755 4627 y Fg(2)1796 4613 y Fi(\))26 b(:=)f(\()p Fl(h)19 b Fk(\001)f Fl(g)2170 4627 y Fg(1)2210 4613 y Fl(;)d(g)2293 4627 y Fg(2)2352 4613 y Fk(\001)j Fl(h)p Fi(\))p Fn(.)41 b(F)-8 b(or)30 b(the)g(pro)s(of)f(of)183 4726 y(the)i(theorem)i(w)m(e)f(wish)f(to)h 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5177 y Fl(;)15 b(h;)g(g)1263 5191 y Fg(2)1304 5177 y Fi(\))26 b Fk(2)f Fi(\001)p Fn(,)k(where)i Fi(\001)e Fn(denotes)i(the)f(diagonal)h(in)f Fl(C)25 b Fk(\002)19 b Fl(C)7 b Fn(.)183 5290 y(With)33 b(this)f(map)i Fl(u)f Fn(w)m(e)g(get)g Fl(h)d Fi(=)f Fl(u)p Fi(\()p Fl(f)5 b(;)15 b(g)s Fi(\))34 b Fn(and)f Fl(h)1878 5257 y Ff(0)1932 5290 y Fi(=)c Fl(u)p Fi(\()p Fl(g)s(;)15 b(f)10 b Fi(\))p Fn(.)49 b(In)32 b(a)h(last)g(step)g(it)g (just)183 5403 y(remains)f(to)h(c)m(hec)m(k)g Fl(h)22 b Fk(\001)g Fl(h)1052 5370 y Ff(0)1104 5403 y Fi(=)28 b Fl(h)1255 5370 y Ff(0)1300 5403 y Fk(\001)22 b Fl(h)29 b Fi(=)f(id)n Fn(,)33 b(whic)m(h)g(can)g(b)s(e)f(guaran)m(teed)i(b)m(y) f(shrinking)p Black Black eop %%Page: 6 6 6 5 bop Black 581 249 a Fn(6)p Black 581 548 a(the)31 b(domains.)722 777 y(So,)f(w)m(e)g(need)g(to)f(c)m(hec)m(k)h(the)g (requiremen)m(ts)g(for)f(applying)h(the)g(implicit)g(function)581 890 y(theorem)e(to)e(the)h(mapping)g Fl(A)p Fn(.)40 b(Therefore,)28 b(w)m(e)f(\034rst)f(need)h(to)g(c)m(hec)m(k)g(the)g(lo)s(cal)f(struc-) 581 1003 y(ture)h(of)f(the)g(in)m(v)m(olv)m(ed)i(spaces)e Fl(C)32 b Fn(and)27 b Fl(D)s Fn(.)39 b(The)26 b(obstructions)h(on)g Fl(D)i Fn(as)c(b)s(eing)i(lo)s(cally)581 1116 y(a)33 b(top)s(ological)i(v)m(ector)e(space)h(are)f(ob)m(viously)h (ful\034lled.)49 b(In)33 b(order)g(to)h(turn)f Fl(C)40 b Fn(in)m(to)34 b(a)581 1229 y(Banac)m(h)27 b(manifold,)g(let)f(us)f (construct)i(a)e(lo)s(cal)h(mo)s(del)g(for)f Fl(C)7 b Fn(.)39 b(First)25 b(it)h(is)f(kno)m(wn)h(that)581 1342 y(the)31 b(space)998 1319 y Fi(~)978 1342 y Fl(C)h Fi(:=)25 b Fl(C)1268 1309 y Fg(1)1307 1342 y Fi(\()p Fl(M)5 b(;)15 b(M)10 b Fi(\))31 b Fn(lies)f(dense)g(in)g Fl(C)36 b Fn(with)31 b(resp)s(ect)f(to)g(the)h Fl(C)3100 1309 y Fg(0)3168 1342 y Fn(-)f(top)s(ology)581 1455 y(and)k(hence)g(pro)m (vides)h(a)e(sk)m(eleton.)51 b(Its)32 b(tangen)m(t)j(space)f(at)g(a)f (p)s(oin)m(t)h Fl(f)40 b Fk(2)3186 1432 y Fi(~)3165 1455 y Fl(C)g Fn(is)33 b(giv)m(en)581 1568 y(b)m(y)28 b(the)g(space)g(of)f (di\033eren)m(tiable)j(sections)e Fi(\000)2128 1535 y Fg(1)2167 1568 y Fi(\()p Fl(f)2257 1535 y Ff(\003)2296 1568 y Fl(T)13 b(M)d Fi(\))p Fn(,)28 b(whic)m(h,)i(of)d(course,)i(serv) m(es)e(as)581 1681 y(a)f(lo)s(cal)g(mo)s(del)g(for)1280 1658 y Fi(~)1259 1681 y Fl(C)7 b Fn(.)38 b(The)27 b(corresp)s(onding)g (lo)s(cal)f(mo)s(del)g(of)g Fl(C)32 b Fn(is)26 b(no)m(w)h(giv)m(en)f(b) m(y)g(the)581 1794 y(space)e(of)e Fh(c)-5 b(ontinuous)24 b Fn(sections)g Fi(\000)1744 1761 y Fg(0)1783 1794 y Fi(\()p Fl(f)1873 1761 y Ff(\003)1912 1794 y Fl(T)13 b(M)d Fi(\))p Fn(.)38 b(It)23 b(inherits)h(the)f(structure)h(of)f(a)g (Banac)m(h)581 1906 y(space,)33 b(if)f(w)m(e)i(de\034ne)f(the)g(norm)g (for)f Fl(X)k Fk(2)29 b Fi(\000)2128 1873 y Fg(0)2167 1906 y Fi(\()p Fl(f)2257 1873 y Ff(\003)2296 1906 y Fl(T)13 b(M)d Fi(\))32 b Fn(b)m(y)h Fk(k)p Fl(X)7 b Fk(k)30 b Fi(:=)f(sup)3119 1928 y Fj(p)p Ff(2)p Fj(M)3295 1906 y Fk(j)p Fl(X)7 b Fi(\()p Fl(p)p Fi(\))p Fk(j)p Fn(,)581 2019 y(where)34 b Fk(j)23 b(\001)g(j)33 b Fn(denotes)h(some)g (arbitrarily)h(c)m(hosen)f(Riemannian)h(metric)f(on)g Fl(M)10 b Fn(.)50 b(Note)581 2132 y(that)35 b(the)g(Banac)m(h)g(space)g (top)s(ology)f(is)g(completely)i(indep)s(enden)m(t)g(of)e(the)g(c)m (hoice)i(of)581 2245 y(the)d(Riemannian)g(metric,)g(since)f(the)h(norm) g(equiv)-5 b(alence)32 b(of)g Fk(j)22 b(\001)f(j)2872 2259 y Fg(1)2944 2245 y Fn(and)32 b Fk(j)22 b(\001)f(j)3239 2259 y Fg(2)3311 2245 y Fn(on)32 b(I)-14 b(R)3523 2209 y Fj(n)581 2358 y Fn(immediately)38 b(carries)f(o)m(v)m(er)g(to)f(an)h (equiv)-5 b(alence)37 b(of)f Fk(k)24 b(\001)h(k)2598 2372 y Fg(1)2673 2358 y Fn(and)37 b Fk(k)25 b(\001)f(k)3019 2372 y Fg(2)3095 2358 y Fn(in)36 b(the)h(sense)581 2471 y(that)31 b Fl(a)p Fk(k)21 b(\001)f(k)981 2485 y Fg(1)1046 2471 y Fk(\024)25 b(k)c(\001)f(k)1298 2485 y Fg(2)1364 2471 y Fk(\024)25 b Fl(b)p Fk(k)20 b(\001)h(k)1655 2485 y Fg(1)1725 2471 y Fn(for)30 b Fl(X)37 b Fn(-)31 b(indep)s(enden)m(t)h Fl(a;)15 b(b)26 b(>)f Fi(0)p Fn(.)722 2700 y(The)38 b(di\033eren)m (tiabilit)m(y)h(condition)g(on)e Fl(A)g Fn(is)f(easily)h(seen,)i(so)e (remains)g(to)h(c)m(hec)m(k)581 2813 y(that)28 b(the)h(F)-8 b(rec)m(het)29 b(deriv)-5 b(ativ)m(e)29 b(of)e Fl(A)h Fn(with)g(resp)s(ect)g(to)g(the)g Fl(C)34 b Fn(-)28 b(co)s(ordinate)h (is)e(in)m(v)m(ert-)581 2926 y(ible)d(in)f(the)g(p)s(oin)m(t)h Fi(\()p Fl(f)5 b(;)15 b Fi(id)o Fl(;)g(f)10 b Fi(\))p Fn(.)38 b(W)-8 b(ell,)25 b(not)f(quite,)h(after)e(a)g(certain)h(pro)5 b(jection,)26 b(whic)m(h)e(is)581 3039 y(due)e(to)g(the)g(fact,)h(that) f(w)m(e)g(just)f(lo)s(ok)h(for)f(solutions)h(of)f Fl(A)g Fn(in)h(the)g(diagonal)h Fi(\001)e Fn(of)g Fl(C)9 b Fk(\002)r Fl(C)e Fn(.)722 3269 y(So)41 b(w)m(e)g(ha)m(v)m(e)h(to)f(pro)m(v)m(e)g (that)h Fl(p)26 b Fk(\001)i Fl(L)40 b Fn(is)g(in)m(v)m(ertible,)45 b(where)d(the)e Fl(C)47 b Fn(-)41 b(deriv)-5 b(ativ)m(e)581 3382 y Fl(L)25 b Fi(=)g Fl(D)839 3396 y Fg(2)879 3382 y Fl(A)p Fi(\()p Fl(f)5 b(;)15 b Fi(id)o Fl(;)g(f)10 b Fi(\))30 b Fn(is)g(giv)m(en)h(b)m(y)1205 3597 y Fl(L)25 b Fi(:)h(\000)1400 3559 y Fg(0)1439 3597 y Fi(\()p Fl(T)13 b(M)d Fi(\))83 b Fk(\000)-15 b(!)83 b Fi(\000)2043 3559 y Fg(0)2082 3597 y Fi(\()p Fl(f)2172 3559 y Ff(\003)2211 3597 y Fl(T)13 b(M)d Fi(\))21 b Fk(\002)f Fi(\000)2579 3559 y Fg(0)2618 3597 y Fi(\()p Fl(f)2708 3559 y Ff(\003)2747 3597 y Fl(T)13 b(M)d Fi(\))1591 3734 y Fl(X)118 b Fk(7!)111 b Fi(\()p Fl(X)28 b Fk(\001)20 b Fl(f)5 b(;)15 b(D)s(f)29 b Fk(\001)21 b Fl(X)7 b Fi(\))581 3950 y Fn(and)27 b Fl(p)e Fi(:)g(\000)931 3917 y Fg(0)971 3950 y Fi(\()p Fl(f)1061 3917 y Ff(\003)1100 3950 y Fl(T)13 b(M)d Fi(\))i Fk(\002)g Fi(\000)1451 3917 y Fg(0)1490 3950 y Fi(\()p Fl(f)1580 3917 y Ff(\003)1619 3950 y Fl(T)h(M)d Fi(\))25 b Fk(!)h Fi([\000)2042 3917 y Fg(0)2081 3950 y Fi(\()p Fl(f)2171 3917 y Ff(\003)2210 3950 y Fl(T)13 b(M)d Fi(\))i Fk(\002)g Fi(\000)2561 3917 y Fg(0)2600 3950 y Fi(\()p Fl(f)2690 3917 y Ff(\003)2730 3950 y Fl(T)h(M)d Fi(\)])p Fl(=)26 b Fk(\030)g Fn(denotes)g(the)581 4062 y(pro)5 b(jection)32 b(on)m(to)f(the)g(diagonal.)722 4292 y(In)f(order)h(to)g (do)f(so,)g(w)m(e)i(\034rst)e(mak)m(e)h(use)f(of)g(the)h(canonical)h (isomorphism)889 4507 y Fl(S)f Fi(:)25 b([\000)1108 4470 y Fg(0)1148 4507 y Fi(\()p Fl(f)1238 4470 y Ff(\003)1277 4507 y Fl(T)13 b(M)d Fi(\))20 b Fk(\002)g Fi(\000)1644 4470 y Fg(0)1683 4507 y Fi(\()p Fl(f)1773 4470 y Ff(\003)1812 4507 y Fl(T)13 b(M)d Fi(\)])p Fl(=)27 b Fk(\030)82 b(\000)-15 b(!)83 b Fi(\000)2548 4470 y Fg(0)2587 4507 y Fi(\()p Fl(T)13 b(M)d Fi(\))1917 4645 y(\()p Fl(X)r(;)15 b(Y)21 b Fi(\))111 b Fk(7!)g Fl(X)27 b Fk(\001)21 b Fl(f)2694 4607 y Ff(\000)p Fg(1)2808 4645 y Fk(\000)f Fl(Y)40 b Fk(\001)20 b Fl(f)3092 4607 y Ff(\000)p Fg(1)3236 4645 y Fl(:)581 4860 y Fn(Multiplying)31 b(with)g Fl(p)20 b Fk(\001)h Fl(L)30 b Fn(on)g(the)h(righ)m(t)g(giv)m(es)g(us)690 5075 y Fi(\()p Fl(S)26 b Fk(\001)20 b Fl(p)g Fk(\001)h Fl(L)p Fi(\)\()p Fl(X)7 b Fi(\))26 b(=)f Fl(S)5 b Fi(\()p Fl(X)28 b Fk(\001)21 b Fl(f)5 b(;)15 b(D)s(f)29 b Fk(\001)20 b Fl(X)7 b Fi(\))26 b(=)f Fl(X)j Fk(\000)20 b Fl(D)s(f)29 b Fk(\001)21 b Fl(X)27 b Fk(\001)21 b Fl(f)2701 5038 y Ff(\000)p Fg(1)2820 5075 y Fi(=)j(\(id)15 b Fk(\000)p Fl(f)3158 5089 y Ff(\003)3196 5075 y Fi(\)\()p Fl(X)7 b Fi(\))52 b Fl(;)581 5290 y Fn(so)30 b(it)h(su\036ces)f(to)g(pro)m(v)m (e)i(that)f Fi(id)14 b Fk(\000)p Fl(f)1847 5304 y Ff(\003)1911 5290 y Fi(:)25 b(\000)2018 5257 y Fg(0)2058 5290 y Fi(\()p Fl(T)13 b(M)d Fi(\))25 b Fk(!)g Fi(\000)2490 5257 y Fg(0)2530 5290 y Fi(\()p Fl(T)13 b(M)d Fi(\))30 b Fn(is)g(an)h(isomorphism.)p Black Black eop %%Page: 7 7 7 6 bop Black 3126 249 a Fn(7)p Black 324 548 a(This)36 b(is)f(the)h(p)s(oin)m(t)g(where)h(the)f(Anoso)m(v)g(prop)s(ert)m(y)g (of)g Fl(f)44 b Fn(comes)36 b(in)m(to)h(pla)m(y)-8 b(.)58 b(By)183 661 y(de\034nition,)28 b(there)f(exists)e(a)h(splitting)g(of)g (the)g(tangen)m(t)i(bundle)e(of)g Fl(M)36 b Fn(in)m(to)27 b(stable)f(and)183 774 y(unstable)33 b(distributions,)h Fl(T)13 b(M)39 b Fi(=)29 b Fl(E)1468 741 y Fj(s)1527 774 y Fk(\002)21 b Fl(E)1691 741 y Fj(u)1736 774 y Fn(.)47 b(Due)33 b(to)g(the)g(fact)f(that)h Fl(D)s(f)42 b Fn(preserv)m(es)183 887 y Fl(E)255 854 y Fj(s)292 887 y Fl(;)15 b(E)404 854 y Fj(u)450 887 y Fn(,)29 b(the)h(pushforw)m(ard)g(of)f Fl(f)38 b Fn(can)30 b(b)s(e)f(split)g(in)m(to)i(t)m(w)m(o)f(mappings)g Fl(f)2650 901 y Fg(0)2715 887 y Fi(:=)25 b Fl(f)2881 901 y Ff(\003)2920 887 y Fk(j)2945 906 y Fg(\000)2989 887 y Fc(0)3024 906 y Fg(\()p Fj(E)3107 887 y Fb(s)3140 906 y Fg(\))183 1000 y Fn(and)36 b Fl(f)409 1014 y Ff(1)519 1000 y Fi(:=)f Fl(f)695 1014 y Ff(\003)734 1000 y Fk(j)759 1019 y Fg(\000)803 1000 y Fc(0)838 1019 y Fg(\()p Fj(E)921 1000 y Fb(u)962 1019 y Fg(\))993 1000 y Fn(.)59 b(By)36 b(the)g(Anoso)m(v)h(prop)s(ert)m(y)g(there)g(exist)f Fl(C)42 b(>)35 b Fi(0)h Fn(and)h Fi(0)f Fl(<)183 1112 y(\025)25 b(<)g Fi(1)i Fn(suc)m(h)g(that)g(for)g(all)g Fl(m)e Fk(2)g Fn(I)-13 b(N)26 b(w)m(e)h(ha)m(v)m(e)h Fk(k)p Fl(f)1817 1080 y Fj(m)1807 1137 y Fg(0)1883 1112 y Fk(k)p Fl(;)15 b Fk(k)p Fl(f)2068 1080 y Ff(\000)p Fj(m)2058 1135 y Ff(1)2190 1112 y Fk(k)26 b Fl(<)f(C)19 b Fk(\001)13 b Fl(\025)2532 1080 y Fj(m)2599 1112 y Fn(.)39 b(Therefore)28 b(w)m(e)183 1225 y(can)i(mak)m(e)i(use)e(of)g(the)h (Neumann)g(metho)s(d)g(for)f(constructing)i(in)m(v)m(erses)f(of)f(b)s (ounded)183 1338 y(linear)h(maps,)f(since)h(the)g(o)s(ccuring)g(series) f(expansions)g(con)m(v)m(erge,)1121 1607 y Fi(\()p Fl(f)1201 1621 y Fg(0)1260 1607 y Fk(\000)20 b Fi(id)o(\))1462 1569 y Ff(\000)p Fg(1)1639 1607 y Fi(=)83 b Fk(\000)1920 1493 y Ff(1)1890 1521 y Fa(X)1879 1715 y Fj(m)p Fg(=0)2047 1607 y Fl(f)2102 1569 y Fj(m)2092 1629 y Fg(0)1056 1909 y Fi(\()p Fl(f)1146 1871 y Ff(\000)p Fg(1)1136 1931 y Ff(1)1260 1909 y Fk(\000)20 b Fi(id)o(\))1462 1871 y Ff(\000)p Fg(1)1639 1909 y Fi(=)83 b Fk(\000)1920 1795 y Ff(1)1890 1822 y Fa(X)1879 2016 y Fj(m)p Fg(=0)2047 1909 y Fl(f)2102 1871 y Ff(\000)p Fj(m)2092 1931 y Ff(1)2273 1909 y Fl(;)183 2201 y Fn(whic)m(h)32 b(pro)m(v)m(es)h(that)f Fl(f)965 2215 y Fg(0)1024 2201 y Fk(\000)21 b Fi(id)30 b Fn(and)i Fl(f)1445 2215 y Ff(1)1540 2201 y Fk(\000)21 b Fi(id)k(=)i Fk(\000)p Fl(f)1948 2215 y Ff(1)2022 2201 y Fi(\()p Fl(f)2112 2168 y Ff(\000)p Fg(1)2102 2223 y Ff(1)2227 2201 y Fk(\000)20 b Fi(id)o(\))32 b Fn(are)f(isomorphisms.) 183 2314 y(This)f(then)h(also)f(holds)h(for)f Fl(f)1196 2328 y Ff(\003)1255 2314 y Fk(\000)20 b Fi(id)29 b Fn(b)m(y)h(the)h (splitting)g(prop)s(ert)m(y)-8 b(.)324 2540 y(So)31 b(w)m(e)g(ha)m(v)m (e)h(sho)m(wn)g(the)f(\034rst)f(part)i(of)e(the)h(pro)s(of)f(of)h(the)g (theorem.)43 b(As)30 b(already)183 2653 y(stated,)d(it)g(remains)g(to)g (sho)m(w)g(that)g(w)m(e)h(can)f(\034nd)g Fl(h;)15 b(h)2028 2620 y Ff(0)2079 2653 y Fn(suc)m(h)27 b(that)g Fl(h)13 b Fk(\001)g Fl(h)2626 2620 y Ff(0)2675 2653 y Fi(=)25 b Fl(h)2823 2620 y Ff(0)2859 2653 y Fk(\001)13 b Fl(h)26 b Fi(=)f(id)n Fn(.)183 2766 y(This)30 b(is)g(done)h(b)m(y)f(simple)h (shrinking)g(of)f(the)h(domains)g Fl(U)2176 2780 y Fg(1)2245 2766 y Fn(and)g Fl(V)2474 2780 y Fg(1)2514 2766 y Fn(.)324 2991 y(Therefore,)j(c)m(ho)s(ose)f(a)f(smaller)h(neigh)m(b)s(orho)s(o)s (d)h Fl(U)2073 3005 y Fg(2)2142 2991 y Fk(\032)28 b Fl(U)2303 3005 y Fg(1)2375 2991 y Fn(of)k Fi(id)f Fn(in)i Fl(C)39 b Fn(suc)m(h)33 b(that)183 3104 y Fl(h)235 3118 y Fg(1)295 3104 y Fk(\001)21 b Fl(h)393 3118 y Fg(2)460 3104 y Fk(2)26 b Fl(U)609 3118 y Fg(1)680 3104 y Fn(for)31 b Fl(h)871 3118 y Fg(1)911 3104 y Fl(;)15 b(h)1003 3118 y Fg(2)1069 3104 y Fk(2)27 b Fl(U)1219 3118 y Fg(2)1258 3104 y Fn(.)43 b(F)-8 b(urther)33 b(adjust)e Fl(V)1983 3118 y Fg(1)2053 3104 y Fn(to)h(the)f(smaller)h(neigh)m(b)s(orho)s(o)s(d)183 3217 y(of)g Fi(id)o Fn(,)i(i.e.,)g(c)m(ho)s(ose)f(a)g(neigh)m(b)s(orho) s(o)s(d)h Fl(V)1587 3231 y Fg(2)1657 3217 y Fk(\032)29 b Fl(V)1810 3231 y Fg(1)1882 3217 y Fn(of)k Fl(f)42 b Fn(suc)m(h)34 b(that)f Fl(u)p Fi(\()p Fl(g)2610 3231 y Fg(1)2650 3217 y Fl(;)15 b(g)2733 3231 y Fg(2)2774 3217 y Fi(\))30 b Fk(2)f Fl(U)2991 3231 y Fg(2)3063 3217 y Fn(for)183 3330 y Fl(g)226 3344 y Fg(1)265 3330 y Fl(;)15 b(g)348 3344 y Fg(2)420 3330 y Fk(2)31 b Fl(V)565 3344 y Fg(2)604 3330 y Fn(.)51 b(Then,)36 b(for)d(an)m(y)i Fl(g)f Fk(2)d Fl(V)1484 3344 y Fg(2)1557 3330 y Fn(w)m(e)k(ha)m(v)m(e)g (that)f Fl(h)g Fn(and)h Fl(h)2423 3297 y Ff(0)2480 3330 y Fn(are)f(in)g Fl(U)2805 3344 y Fg(2)2878 3330 y Fn(\()p Fl(f)44 b Fn(is)33 b(in)183 3443 y Fl(V)236 3457 y Fg(2)306 3443 y Fn(an)m(yw)m(a)m(y\))f(and)f(therefore,)h(b)m(y)e(de\034nition,) j Fl(h)21 b Fk(\001)f Fl(h)1969 3410 y Ff(0)1993 3443 y Fl(;)15 b(h)2085 3410 y Ff(0)2129 3443 y Fk(\001)21 b Fl(h)26 b Fk(2)f Fl(U)2401 3457 y Fg(1)2441 3443 y Fn(.)41 b(Hence,)31 b Fl(h)21 b Fk(\001)g Fl(h)2972 3410 y Ff(0)3026 3443 y Fn(and)183 3556 y Fl(h)235 3523 y Ff(0)279 3556 y Fk(\001)f Fl(h)30 b Fn(are)h(the)g(unique)g(solutions)f (of)893 3760 y Fi(\()p Fl(h)21 b Fk(\001)f Fl(h)1098 3723 y Ff(0)1122 3760 y Fi(\))g Fk(\001)h Fl(g)54 b Fi(=)c Fl(h)21 b Fk(\001)f Fl(f)30 b Fk(\001)20 b Fl(h)1731 3723 y Ff(0)1805 3760 y Fi(=)50 b Fl(g)24 b Fk(\001)d Fi(\()p Fl(h)g Fk(\001)f Fl(h)2244 3723 y Ff(0)2268 3760 y Fi(\))885 3898 y(\()p Fl(h)972 3860 y Ff(0)1016 3898 y Fk(\001)h Fl(h)p Fi(\))g Fk(\001)f Fl(f)60 b Fi(=)50 b Fl(h)1493 3860 y Ff(0)1537 3898 y Fk(\001)20 b Fl(g)k Fk(\001)d Fl(h)51 b Fi(=)f Fl(f)29 b Fk(\001)21 b Fi(\()p Fl(h)2126 3860 y Ff(0)2170 3898 y Fk(\001)f Fl(h)p Fi(\))183 4102 y Fn(in)30 b Fl(U)350 4116 y Fg(1)420 4102 y Fn(and)h(therefore)g (equal)f(to)h(the)f(iden)m(tit)m(y)j(map.)41 b(This)30 b(pro)m(v)m(es)h(the)g(theorem.)p Black Black eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF