数理解析研究所講究録
1118 巻 1999 年 13-26

13

Stable and unstable manifolds ofdiffeomorphisms with positive entropy
Naoya Sumi (鷲見 直哉)
Abstract
We show that -diffeomorphisms with positive entropy are chaotic
in the sense of Li-Yorke. To do so we prove that these maps arechaotic on the closure of stable manifolds for some points. The notionof $”*$ -chaos” was introduced by Kato and it is related to chaos in thesense of Li-Yorke.
$C^{2}$$*-$

1

Introduction

We study chaotic properties of diffeomorphisms with positive entropy. Notionsof chaos have been given by Li and Yorke [9], Devaney [2] and others. It is
known that if a continuous map of interval has positive entropy, then it ischaotic according to the definition of Li and Yorke (cf. [1]).
In [6] Katok proved the following :let be a-diffeomorphism of aclosed surface. If the topological entropy of is positive, then there exists a
hyperbolic set such that the restriction of into is topologically conjugate
to a subshift of finite type with positive entropy. This implies that is chaotic
in the sense of Li-Yorke.
However, Katok’s theorem does not hold for the high dimensional case.
Indeed, let be a surface diffeomorphism with positive entropy and let :be an irrational rotation. Then a product map
has the same
positive entropy, but it does not have as above because there are no periodic
points of.
In this paper we show the following:$C^{1+\epsilon}$

$f$

$f$

$\Gamma$$f$$\Gamma$

$f$

$f$

$r$

$S^{1}arrow S^{1}$$f\cross r$$\Gamma$$f\cross r$

Theorem A Let be a -diffeomorphism of a closed-manifold. If the
topological entropy of is positive, then is chaotic in the sense of Li-Yorke.
$C^{2}$$f$$f$$C^{\infty}$$f$

To my knowledge this theorem gives the most simplest sufficient condition
for chaotic phenomena of high dimensional dynamical systems. It remains
a question whether Theorem A is true for homeomorphisms. However thisquestion is still unsolved.
Let $M$ be a closed-manifold and let be the distance for $M$ induced
by a Riemannian metric
on $M$ . A subset of $M$ is a scrambled set of
if there is a positive number $\tau>0$ such that for any $y\in S$ with $x\neq y$ ,$C^{\infty}$$||\cdot||$

$d$

$S$

$f$

$x,$

14

1. $\lim\sup_{narrow\infty}d(f^{n}(X), fn(y))>\tau$ ,2. $\lim\inf_{narrow\infty}d(fn(x), fn(y))=0$.
If there is an uncountable scrambled set of , then we say that is chao
in the sense of $Li$-Yorke. Li and Yorke showed in [9] that if $f:[0,1]arrow[0,1]$ is, then is chaotic in thisa continuous map with a periodic point of: for any $x\in S$ andsense. In [9] there was the following one more
any periodic point $p\in M,$ $\lim\sup_{narrow\infty}d(f^{n}(X), fn(p))>0$ . But this conditionis unnecessary because a scrambled set contains at most one point which does
not satisfy this condition. For the examples and the properties of scrambled
sets, the readers may refer to [1], [5], [11], [12], [13], [22], [23] and [24].
Concerning the chaos in the sense of Li-Yorke, Kato introduced the notionof $”*$ -chaos” as follows: let $F$ be a closed subset of $M$ . A map : $Marrow M$
is -chaotic on $F$ (in the sence of Li-Yorke) if the foilowing conditions are
satisfied:
$S$

$f$

$f$

$\mathrm{t}ic$$f$$\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{i}_{0}\mathrm{d}3$$\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{t}\mathrm{i}’ \mathrm{o}\mathrm{n}$

$f$

$*$

1. there is $\tau>0$ such that if and are any nonempty open subsets of $F$
with
and $N$ is any natural number, there is a natural number$n\geq N$ such that $d(f^{n}(x), fn(y))>\tau$ for some $x\in U,$ $y\in V$ , and
$V$

$U$

$U\cap V=\emptyset$

2. for any nonempty open subsets $U,$
of $F$ and any
there is anatural number $n\geq 0$ such that $d(f^{n}(x), fn(y))<\epsilon$ for some $x\in U$ ,$\epsilon>0$$V$.

$y\in V$

Such a set, isof$S,\overline{S}$$F$

is called-chaotic set. If is a scrambled set, then the closure-chaotic set. In [4] Kato showed that the converse is true.
$S$$\mathrm{a}*$$\mathrm{a}*$Lemma 1.1 ([4], Theorem 2.4) If : $Marrow M$ is continuous and ischaotic on $F$ , then there -set $S\subset F$ such that is a scrambled set of and. If $F$ is perfect ($i.e$ . has no isolated points), we can choosethat it is a countable union of Cantor sets.
$f$$\overline{S}=F$

$*-$

$S$

$F_{\sigma}$

$f$

$F$

$S\mathit{8}uch$

To obtain Theorem A we need the following theorem.
Theorem
Let be a -diffeomorphism of a closed-manifold $M$ andlet be an -invariant ergodic Borel probability measure on M. If the metric
entropy of is positive, then for -almost all $x\in M$ the following hold:$\mathrm{B}$

$\mu$

$f$

$C^{2}$$C^{\infty}$

$f$

$\mu$

$\mu$

$(a)\overline{W^{s}(x)}$is a $perfeci*$ -chaotic set; and$(b)\overline{W^{u}(x)}$contains a $perfeCt*$ -chaotic set.

Here $W^{s}(x)$ and $W^{u}(x)$ are defined by$W^{s}(X)= \{y\in M:\lim\sup \mathrm{l}\mathrm{o}\underline{1}\mathrm{g}d(fn(x), fn(y))<0\}$$narrow\infty n$and,$W^{u}(X)= \{y\in M:\lim\sup\log\underline{1}d(f^{-n}(x), f^{-}n(y))<0\}$$narrow\infty$

respectively.

$n$

15

We notice that for -almost all $x\in M$ , the above sets $W^{s}(x)$ and $W^{u}(x)$ are
immersed manifolds under the assumptions of theorem B. Indeed, let andbe as above. For -almost all $x\in M$ , there exist a splitting of the tangent
space $T_{x}M=\oplus_{i=}Es(x_{1i})(x)$ and real numbers $\lambda_{1}(x)<\lambda_{2}(x)<\cdots<\lambda_{s(x)(x})$
such that$\mu$$C^{2}$

$f$

$\mu$

$\mu$

$(a)$

the maps $xrightarrow E_{i}(x),$and $s(x)$ are Borel measurable, moreover$s(x)$ are-invariant $(i=1, \cdots, s(x))$ ,$\lambda_{i}(x)$$E_{i}(f(x))=D_{x}f(Ei(x))$ and$\lambda_{i}(X),$$(b) \lim_{narrow\pm\infty}\frac{1}{n}\log||D_{x}f^{n}(v)||=\lambda_{i}(X)$$f$$(0\neq v\in E_{i}(x), i=1, \cdots , s(x))$and$(c) \lim_{narrow\pm\infty}\frac{1}{n}\log|\det(Df^{n}x)|=\sum^{s}\lambda_{i}(i=1(x)x)\dim Ei(x)$([14]). The numbers
are called Lyapunov exponents ofat . Since is ergodic, we can put $s=s(x),$and $m_{i}=\dim E_{i}(x)$$(i=1, \cdots , s)$ for$x\inM$-almost all.
Letdenote the metric entropy of (see [10] for definition). A wellknown theorem of Margulis and Ruelle [21] says that entropy is always bounded
above by the sum of positive Lyapunov exponents; i.e.
.
Since has positive entropy, we have$\lambda_{1}(x),$$x$$\cdots,$ $\lambda_{s(}x)(X)$

$f$

$\mu$

$\lambda_{i}=\lambda_{i}(x)$$\mu$$h_{\mu}(f)$$f$$h_{\mu}(f)\leq\Sigma_{\lambda_{i}>0i}\lambda m_{i}$$f$$0<h_{\mu}(f) \leq\max\{\lambda_{i}\}=\lambda s$.

Therefore, by Pesin’s stable manifold theorem ([3], [15], [17]), the set $W^{u}(x)$is the image of a
injective immersion of an euclidean space such that
$T_{x}W^{u}(x)=\oplus_{\lambda_{i>}}0E_{i}(x)(\neq\{0\})$for -almost all $x\in M$ . $W^{u}(x)$ is called an unstable manifold. Similarly, $W^{s}(x)$ is a
immersed manifold because $W^{s}(x)$ is
the unstable manifold of, which has positive entropy $h_{\mu}(f^{-1})=h(\mu f)>0$ .
$W^{s}(x)$ is called a stablemanifold.
Let us see how Theorem A follows from Theorem B. We denote as $h(f)$the topological entropy of(see [10] for definition). Then we know that$h(f)= \sup\{h_{\mu}(f) : \mu\in \mathcal{M}_{e}(f)\}$ whereis the set of all -invariant
ergodic Borel probability measures (cf [20]). Thus, if $h(f)>0$ , then we canchoosewith $h_{\mu}(f)>0$ . Therefore, by Theorem and Lemma 1.1,
is chaotic in the sense of Li-Yorke.
Remark that (a) of Theorem does not hold for unstable manifolds ingeneral. For example, the Smale horseshoe has unstable manifolds which intersect a stable manifold of a fixed point. Since all points in the stablemanifold converge to the fixed point, they do not satisfy the first condition of-chaos.
Now we shall give a sufficient condition that is -chaotic onas$C^{2}$$\mu$$C^{2}$$f^{-1}$$f$$\mathcal{M}_{e}(f)$$\mathrm{f}$$\mu\in \mathcal{M}_{e}(f)$$\mathrm{B}$$f$$\mathrm{B}$$(\mathrm{c}\mathrm{f}.[18])$

$*$

$f$

$\overline{W^{u}(x)}$$*$follows.

TheoremIf

an ergodic $SRB$ measure; then bothare $perfeCt*$ -chaotic sets for -almost all $x\in M$ .
$\mathrm{C}$$\mu i\mathit{8}$$\overline{W^{s}(x)}$and$\overline{W^{u}(X)}$$\mu$

If is a measurable decomposition of $M$ , then a familyofBorel probability measures exists, and it satisfies the following conditions:$\xi$$\{\mu_{x}^{\xi}|x\in M\}$

16

1. for $x,$ $y\in M$ if $\xi(x)=\xi(y)$ thencontaining in ,$x$2.

$\mu_{x}^{\xi}(\xi(x))=1$$\mu_{x}^{\xi}=\mu_{y}^{\xi}$, here$\xi(x)$denotes a set$\xi$

for -almost all $x\in M$ ,$\mu$

3. for any Borel set.

$A$a function$xarrow\#\mu_{x}^{\xi}(A)$is measurable and $\mu(A)=$$\int_{M}\mu_{x}^{\xi}(A)d\mu(X)$The family
is called a canonical system of conditional measures
for and (see [19] for more details).
An -invariant Borel probability measure is called a Sinai-Ruelle-Bowen
measure (SRB measure for abbreviation) provided$\{\mu_{x}^{\xi}|x\in M\}$

$\xi$

$\mu$

$f$

$(A)$

$\mu$

for -almost all $x\in M$ , there exists a positive Lyapunov exponent of ,
$x$

$\mu$

has a conditional measure that is absolutely continuous (with respect
to the Lebesgue measure) on unstable manifolds, which is defined asfollows:$(B)\mu$

From (A), the unstable manifold $W^{u}(x)$ is asubmanifold for -almost
all in $M$ . Let
denote the Lebesgue measure of $W^{u}(x)$ . A measurable
decomposition of $M$ is said to be subordinate to unstable manifolds if for-almost all in $M$$C^{2}$

$\mu$

$x$

$m_{x}^{u}$

$\xi$

$x$

$\mu$

$(C)\xi(x)\subset W^{u}(x)$$(D)\xi(x)$,contains an open neighborhood of$x$

in $W^{u}(x)$ .

We say that has an absolutely continuous conditional measure on unstablemanifolds provided is absolutely continuous with respect tofor -almost
all in $M$ if is subordinate to unstable manifolds. It is known ([7], [8]) thatsatisfies (B) if and only if the following equation holds:$\mu$$\mu_{x}^{\xi}$$m_{x}^{u}$

$x$

$\mu$

$\xi$

$\mu$

$h_{\mu}(f)= \int_{\lambda i(x}\sum_{>)0}\lambda_{i}(x)\dim E_{i}(x)d\mu(X)$.

This is sometimes known as Pesin’s formula. For the examples and thestochastic properties of diffeomorphisms with SRB measures, the readers may
refer to [25]. In a similar way we cana measurable partition subordinate
to stable manifolds.

defin.e2Preliminaries

In this section we introduce -invariant measurable partitions each of whose
elements is contained in the closure of (un)stable manifolds. Let be adiffeomorphism of a closed-manifold $M$ and let be an -invariant ergodic
Borel probability measure on $M$ with $h_{\mu}(f)>0$ . Denote as the family of
Borel sets.
$f$

$f$

$C^{\infty}$

$\mu$

$f$

$B$

$C^{2}-$

17

Lemma 2.1 (Proposition 3.1 [7]) Let and be as above. Then there
measurable partitions
andof $M$ such that$f$$exi_{\mathit{8}}t$$(a)\xi^{s}\leq f\xi^{s}$and$\xi^{u}\leq f^{-1}\xi^{u}$$\mu$$\xi^{u}$$\xi^{s}$,andare subordinate to stable manifolds and unstable manifolds
respectively,$(b)\xi^{s}$$\xi^{u}$

$(c)$

both

$\mathrm{V}_{n=0}^{\infty}fn\xi^{s}$

$(d)$

for

-almost all $x\in M$ ,

$\mu$

and

$\mathrm{v}_{n=0}^{\infty}f-n\xi^{u}$are the partitions into points,

and$\bigcup_{n=0}^{\infty}f^{-n}\xi S(f^{n}(x))=W^{s}(x)$Lemma 2.2 (Corollary 5.3 [8]) Let
be as in Lemma 2.1. Then,$f$$\bigcup_{n=0}^{\infty}f^{n}\xi u(f-n(x))=W^{u}(x)$

and

$\mu$

be as above and let.

$\xi^{\sigma}(\sigma=$$s,$ $u)$$h_{\mu}(f)$$=H_{\mu}(f \xi s|\xi^{s})=\int-\log\mu x(f\xi s(X)\xi^{s})d\mu(X)$$=H_{\mu}(f^{-}1 \xi u|\xi u)=\int-\log\mu^{\xi^{u}}x(f^{-}1\xi^{u}(X))d\mu(x)$where the family
for and .

$\{\mu_{x}^{\xi^{\sigma}}|x\in M\}$is a canonical system of conditional measures$\xi^{\sigma}$$\mu$

Let us introduce two measurable partitions defined by$\eta^{s}=\bigwedge_{i=0}^{\infty}f^{-}i\xi^{s}$and$\eta^{u}=\bigwedge_{i=0}^{\infty}f^{i}\xi u$.

By Lemma 2.1(a) and (d) we can easily check thatandfor -almost all $x(\sigma=s, u)$ . Let
be a canonical systemof conditional measures for and. By Doob’s theorem it follows
that for$f\eta^{\sigma}=\eta^{\sigma}$$\overline{W^{\sigma}(x)}$$\eta^{\sigma}(x)\subset$$\{\mu_{x}^{\sigma}|x\in M\}$$\mu$$\eta^{\sigma}(\sigma=s, u)$$\mu$$\mathrm{A}\in B$$\mu_{x}^{s}(A)=\lim_{narrow\infty}\mu_{x}^{j\xi}-hS(A)$and( -almost all ).

$\mu_{x}^{u}(A)=\lim_{narrow\infty}\mu_{x}^{f\xi}(nuA)$

$x$

$\mu$

Since
and preserves , we have( -almost all )forand $\sigma=s,$ .
Let $C(M)$ be the Banach space of continuous real-valued functions of $M$, and letwith the $\sup$ norm
be a set of all Borel probability$M$measures on with the weak topology. Since $C(M)$ is separable, there exists
a countable set
which is dense in $C(M)$ . For
define$f\eta^{\sigma}=\eta^{\sigma}$$\mathrm{A}\in B$$f$$\mu_{x}^{\sigma}(A)=\mu_{fx}^{\sigma}(fA)$$\mu$$||\cdot||$$\mathcal{M}(M)$$\{\varphi_{1}, \varphi_{2}, \cdots\}$$\nu,$$\rho(\nu, \nu’)=n1\sum_{=}\frac{|\int\varphi_{n}d\nu-J\varphi_{n}d\nu’|}{2^{n}||\varphi_{n}||}\infty$Then$\rho$

$x$

$\mu$

$u$

is a compatible metric for$\mathcal{M}(X)$and$\nu’\in \mathcal{M}(X)$.

$(\mathcal{M}(X), \rho)$

is compact$(\mathrm{c}\mathrm{f}.[10])$

.

18

and
Lemma 2.3 Let $f,$$\sigma=s,$
there exists a closed set

be$\{\mu_{x}^{\sigma}|x\in M\}$

$\mu$

$u$

$F_{\epsilon}^{\sigma}$

with$a\mathit{8}$andabove. Then forsatisfying the map$\epsilon>0$$\mu(F^{\sigma})\in\geq 1-\epsilon$$F_{\epsilon}^{\sigma}\ni xrightarrow\mu_{x}^{\sigma}\in \mathcal{M}(X)$$i\mathit{8}$continuous.

be as above. By the definition of conditional

Proof. Let
measures the map$\{\varphi_{1}, \varphi_{2}, \cdots\}$$M \ni x-\rangle\int\varphi_{n}d\mu_{x}\sigma$

is Borel measurable for $n\geq 1$ . Thus, by Lusin’s theorem, for $n\geq 1$ therewithexists a closed set
satisfying$\mu(F_{n}^{\sigma})\geq 1-\epsilon/2^{n}$$F_{n}^{\sigma}$: continuous.

$F_{n}^{\sigma} \ni x\vdash+\int\varphi_{n}d\mu_{x}^{\sigma}$

Then$F_{\epsilon}^{\sigma}= \bigcap_{n=1}^{\infty}F_{n}^{\sigma}$has the desired property.
$\square$Lemma 2.4 Let $f,$$x\in M$ and $\sigma=s,$ $u,$

$\mu$

and

be as above. Then for -almost all
has no isolated points.

$\{\mu_{x}^{\sigma}|x\in M\}$$\mathit{8}upp(\mu_{x})\sigma$$\mu$

Proof. We will give the proof for $\sigma=u$ and so we here omit for $\sigma=s$since the technique of the proof is similar.
If this lemma is false,
has an isolated point for belonging topositivesome Borel set withmeasure. Let
be as in Lemma 2.1. Since
diam $(f^{-k}\xi^{u}(X))arrow 0(karrow\infty)$ by Lemma 2.1 (c),$x$$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\mu_{x}^{u})$$\xi^{u}${$P_{-k}=$$x\in M:\mu_{x}^{f^{-k}}\xi^{u}$is a point measure}has positive -measure for large enough. Since is -invariant, we havefor -almost all and. Then$k$$\mu$$f_{*}^{n}\mu_{x}f^{-ku}\xi=\mu_{f^{n}x}^{f\xi}n-ku$

$=$

$=$

$=$

{
{
{

$f^{n}(x)\in M:\mu_{x}^{f^{-k}\xi^{u}}$$x\in M:f_{*}n\mu_{fx}^{f}--nk_{\xi^{u}}$$(n\in \mathbb{Z})$is a point measure}is a point measure}is a point measure}$x\in M:\mu_{x}^{f^{n-ku}}\xi$$=P_{n-k}$$f$$n\in \mathbb{Z}$

$x$

$\mu$

$f^{n}(P_{-k})$

$\mu$

.

Put
$P= \bigcap_{j\geq\geq jn-}1^{\cup}nPk=\mathrm{n}_{j\geq}1\cup n\geq jf^{n}P_{-}k$.

is -invariant and is ergodic, we have $\mu(P)=1$ .
For
there exists an increasing sequencesuch thatfor$i\geq 0$ . Since
andis a point measure
for , so is . Therefore
is a point measure for -almost all $x\in M$ .

Since

$P$

$f$

$\mu$

$x\in P$$\{n_{i}\}_{i>0}$$\mu_{x}^{u}=\lim_{narrow\infty}\mu_{x}!^{n}\xi^{u}=\lim_{iarrow\infty}\mu_{x}^{f^{n}}i\xi^{u}$$i$$\mu_{x}^{u}$$\mu_{x}^{u}$$\mu_{x}^{\overline{f}^{n_{i}}\xi^{u}}$$\mu$$x\in P_{n_{i}}$19Since is finer thanso is . Thus$\xi^{u}$$\eta^{u}$and$\mu_{x}^{u}$$\mu_{x}^{\xi^{u}}(f-1\xi^{u}(x))=1$$\mu_{x}^{\xi^{u}}$is a point measure for -almost all $x\in M$ ,for -almost all . Therefore
$\mu$

$x$

$\mu$

$h_{\mu}(f)= \int-\log\mu_{x}^{\xi^{u}}(f^{-}1\xi u(x))d\mu(_{X})=0$

by Lemma 2.2. This is a contradiction.
$\square$

Let $B(x, r)$ and $U(x, r)$ denote the closed and open balls in $M$ with center$x\in M$ and radius $r>0$ respectively.

Lemma 2.5 Let $f,$

$\mu$

and

$\{\mu_{x}^{\sigma}|x\in M\}$$x\in M$($\overline{W^{s}(y)}=\overline{W^{s}(x)}$Proof. Let$\xi^{s}$$\mu_{x}^{s}$be as above. Then for -almost all$\mu$-almost all $y\in M$ ).

be as in Lemma 2.1. Then we have that for -almost all$\mu$$x\in M$$\xi^{s}(y)\subset\overline{W^{s}(x)}$($\mu_{x}^{s}$-almost all ).
$y$Indeed, letdenote the distance induced by the Riemannian metric on$W^{s}(x)$ .
Then there exist an increasing familyof closed sets of $M$
withand a sequenceof positive numbers satisfying$d_{x}^{s}$$\{\Lambda_{l}\}_{l\geq 1}$$\mu(\bigcup_{\ell\geq}1\Lambda\ell)=1$$\{A_{l}\}_{\ell\geq 1}$

that(e) for each$x\in\Lambda_{l}$there exists $\epsilon=\epsilon(x)>0$ , such that

$B(x, \epsilon)\mathrm{n}\Lambda_{l}\ni y\text{ト}arrow W_{A}^{\mathit{8}}(ty)=\{Z\in W^{s}(y) :is continuous with respect to the Hausdorff metric$\Lambda_{\ell\ni}\lim_{xyarrow}dH(W_{A}^{S}t(y), WA_{\ell}s(X))=0$(f) for each$x\in\Lambda_{l},$$(\mathrm{C}\mathrm{f}.[7], [15], [16])$$d_{H}$: i.e.

,$\xi^{s}(X)\subset W_{A_{t}}^{s}(x)$. Take arbitrary

be as in (e) and letd^{s}x(z, y)\leq A_{l}\}$$0<r<\epsilon$$\mu_{x}^{s}(\bigcup_{t\geq 1\ell}\Lambda)=1$$y\in \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\mu_{x}^{s}|\Lambda_{\ell})(\ell\geq 1)$. Let $\epsilon=\epsilon(y)>0$. Recall that for -almost all $x\in M$
$\mu$

and

$(\ell\geq 1)$$\mu_{x}^{s}|\Lambda_{\ell}=\lim_{narrow\infty}\mu_{x}f^{-}n\xi^{\mathit{8}}|\Lambda_{\ell}$.

Since $U(y, r)$ is open, we havefor large enough. So
we can take $y’\in U(y, r)\cap\Lambda_{l}\cap f^{-n}\xi^{S}(x)$ . Since $y’\in f^{-n}\xi^{s}(x)\subset W^{s}(x)$ , wehave
. Since $y’\in U(y, r)\cap\Lambda_{l}$ and is arbitrary, it follows
that$\mu_{x}^{f^{-n}\xi^{s}}(U(y, r)\cap\Lambda_{\ell})>0$$W_{A_{t}}^{\mathit{8}}(y^{;})\subset W^{s}(x)$

$n$

$r$

.

$\lim_{rarrow 0}d_{H}(W_{A_{t}}^{s}(y’), W_{A_{\mathit{1}}}^{s}(y))=0$

Therefore.
From this fact it follows that for $n\geq 0$ and -almost all $x\in M$$\xi^{s}(y)\subset W_{A_{\ell}}^{s}(y)\subset\overline{W^{s}(x)}$$\mu$$\xi^{s}(f^{n}(y))\subset\overline{W^{s}(f^{n}(x))}$($\mu_{x}^{s}$-almost all ).
$y$

20

Thus
$W^{s}(y)= \bigcup_{n\geq 0}f^{-n}\xi s(f^{n}y)\subset\bigcup_{n\geq 0}f^{-n}(\overline{Ws(f^{n(_{X)}})})\subset\overline{W^{s}(X)}$for -almost allmeasures,-almost all .
$\mu_{x}^{s}$$y$$\mu_{y}^{s}=\mu_{x}^{s}$$\overline{W^{s}(y)}=\overline{W^{s}(x)}\mathrm{f}\mathrm{o}\mathrm{r}\square$$y$$\mu_{x}^{\mathit{8}}$3. On the other hand, by the definition of conditional
for $y\in\eta^{s}(x)$ . This implies that

Proof of Theorem$\mathrm{B}(\mathrm{a})$. Let be a
The purpose of this section is to show Theorem$M$phism of a closed-manifold
and let be an -invariant ergodic Borel
probability measure on $M$ with positive entropy. As described in \S 1 the stable
immersed manifold for -almost all $x\in M$ and so the
manifold $W^{s}(x)$ is a$W^{s}(x),$, is perfect.
closure ofLetandbe as in \S 2. By Lemma 2.4,
has noisolated points for -almost all $x\in M$ . Therefore, to obtain the conclusion it
suffices to show the following.
$C^{\infty}$$C^{2_{-}}\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{e}\mathrm{o}\mathrm{m}\mathrm{o}\mathrm{r}_{-}$$f$$\mathrm{B}(\mathrm{a})$

$f$

$\mu$

$C^{2}$$\mu$$\overline{W^{s}(X)}$$\{\mu_{x}^{s}|x\in M\}$$\eta^{s}$$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\mu_{x}^{s})$$\mu$Proposition 1 If
not a point measure for -almost all $x\in M$ , then$a*$is-chaotic set for -almost all $x\in M$ .
$\mu_{x}^{s}i\mathit{8}$$\overline{W^{s}(x)}$

$\mu$

$\mu$

be as in Lemma 2.3. By assumption
Proof. Fix $0<\epsilon<1$ and let
we can take and fix
such thatis not a point measure.
is not one point, there are disjoint open sets
andof $M$
Since
satiswing that
$F_{\epsilon}^{s}$$x_{0}\in \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\mu|F_{\mathrm{g}}^{s})$$\mu_{x_{0}}^{s}$$O_{1}$$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\mu_{x}^{s}0)$$d(O_{1}, \mathit{0}_{2})=\inf\{d(x, y) :$\mu_{x_{0}}^{s}(Oi)>\delta$

x\in O_{1}, y\in O_{2}\}>\delta$$x\in U(x_{0},6’)\cap F_{\epsilon}^{s}$. Put$\epsilon’>0$

such that(2)$K= \bigcap_{n=0}^{\infty}\bigcup_{k\geq n}f^{-k}(U(X0, \epsilon’)\cap F_{\epsilon}^{s})$, by ergodicity ofTake arbitrary within Lemma 2.1 and put$\delta’$(1)$(i=1,2)$$\mu_{x}^{s}(O_{i})>\delta$$\mu(U(x_{0}, \epsilon’)\cap F_{\epsilon}^{s})>0$

and$(i=1,2)$

for some $\delta>0$ . By Lemma 2.3 we can choosefor$O_{2}$$\mu$

we have $\mu(K)=1$ .

$0< \sqrt{\delta’}<\min\{\mu(U(x_{0,\mathcal{E})}’\cap F_{\epsilon}^{s}), \delta\}$

. Let

.
$\xi^{s}$

Since
be as

(3)

$A_{m}^{s}(n)=\{x\in M|d_{H}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}((f^{-}[k/2]\xi^{s}(f^{[/2}k]X),\overline{W^{S}(_{X)}}fk-[k/2]\xi s(f[k/2]x))\leq 1/m()\leq 1/mk’\geq n)\}$for $m\geq 1$ . Thenand
2.1 (c) and (d). Thus there exists an increasing sequence$\mu(A_{m}^{s}(n_{m}))\geq 1-\delta’/2^{m}(m\geq 1)$ . Since$n,$$A_{m}^{S}(n)\subset \mathrm{A}_{m}^{s}(n+1)$$\mu(\bigcup_{n=0}^{\infty}A_{m}^{S}(n))=1$$\{n_{m}\}$by Lemma
such that$\int\mu_{x}^{s}(\bigcap_{m}\infty \mathrm{A}^{s}(n_{m}))d\mu=\mu(\bigcap_{m1m}^{\infty}A^{s}(n_{m}))\geq 1-\sum_{m=1}^{\infty}\delta’/=1m=12^{m}=-\delta’$,21

we can find a Borel set$C_{\delta}^{S},$$\subset M$with$\mu(o_{\delta}^{S},)\geq 1-\sqrt{\delta’}$satisfying(4)$\mu_{x}^{s}(\bigcap_{m1m}\infty=A^{s}(n_{m}))\geq 1-\sqrt{\delta^{J}}$for. To obtain the conclusion it sufficies to show thatis a, because is arbitrary.
-chaotic set forFor, by the definition of $K$ , there exists a sequence of positive
integers
with $k_{m}>n_{m}$ such that
. Then by (4)and (2)$x\in C_{\delta}^{s}$$\overline{W^{s}(x)}$$*$$x\in K\cap C_{\delta}^{S}$$\delta’$$x\in K\cap C_{\delta}^{s},$$\{k_{m}\}_{m}$$f^{k_{m}}(x)\in U(x_{0}, \mathcal{E}’)\cap F_{\epsilon}^{s}$,$\mu_{x}^{s}(A_{m}^{s}(k_{m}))\geq\mu_{x}^{s}(\bigcap_{m}\infty=1mA^{s}(k_{m}))\geq\mu_{x}S(\mathrm{n}_{m=}^{\infty}1mA^{S}(n_{m}))\geq 1-\sqrt{\delta’}$$(i=1,2, m\geq 1)$ .

$\mu_{x}^{s}(f^{-k}m(O_{i}))=\mu_{j}^{s}k_{m}(x)(Oi)>\delta>\sqrt{\delta’}$Thus we have $\mu_{x}^{s}(A_{m}^{S}(k_{m})\cap f^{-k_{m}}(o_{i}))>0$ for $i=1,2$ and $m\geq 1$ . From
Lemma 2.5 we may assume that for $m\geq 1$ and $i=1,2$ there exists a point$y_{i}=y_{i}(m)\in A_{m}^{s}(k_{m})\cap f^{-k_{m}}(O_{i})$ suchthat$\overline{W^{s}(y_{i})}=\overline{W^{S}(X)}$.

By (3) we have,$d_{H}(f^{-[k_{m}}/2]\xi s(f^{[}km/2]yi),\overline{W^{s}(X)})=d_{H}(f^{-}[km/2]\xi S(f^{[}km/2]y_{i}),\overline{Ws(y_{i})})\leq 1/m$

diam $(f^{k_{m^{-}}}[km/2]\xi s(f^{[}km/2]yi))\leq 1/m$

$U_{1}$To show thatandsatisfy$\overline{W^{S}(x)}$is$\mathrm{a}*$$(i=1,2, m\geq 1)$.

(5)-chaotic set, suppose that nonempty open sets$U_{2}$$U_{1^{\cap}}U2\neq\emptyset$,$U_{j}\cap\overline{Ws(_{X})}\neq\emptyset$$(j=1,2)$ .

By (5) we may assume that$y_{i}\in f^{-}[km/2]\xi s(f^{[}km/2]yi)\cap U_{j}\neq\emptyset$,$f^{k_{m}}(y_{i})\in f^{k_{m}-[k/2]}m\xi S(f^{[k}m/2]iy)\subset O_{i}$$(1\leq i,j\leq 2)$(6)

if $m$ is sufficiently large. Take$a_{i,j}=a_{i,j(m})\in f^{-[/}k_{m}2]\xi^{s}(f[km/2]yi)\cap U_{j}$for $1\leq i,j\leq 2$ . Then we have that for $1\leq i,j\leq 2$$a_{i,j}\in U_{j}$,$d(f^{k_{m}}(a1,1),$ $f^{k}m(a2,2))>\tau$

and $d(f^{k_{m}}(a1,1),$ $f^{k}m(a1,2))<1/m$by (1), (5) and (6). Since $m$ is arbitrary,$K\cap C_{\delta}^{s},$.

$\overline{W^{s}(x)}$is a -chaotic set for $x\in$$*$$\square$22

4

Proof of Theorem$\mathrm{B}(\mathrm{b})$In this section we will prove Theorem. Let $f,$andbe as in \S 2. By Lemma 2.4,
has no isolated points for -almost all$x\in M$ . Therefore, to obtain the conclusion it sufficies to show the following.
$\mathrm{B}(\mathrm{b})$$\mu,$$\{\mu_{x}^{u}|x\in M\}$$\eta^{\mathrm{u}}$$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\mu_{x}^{u})$$\mu$Proposition 2 If
is not a point measure for -almost all $x\in M$ , thensupp
is $a*$ -chaotic set for
all $x\in M$ .
$\mu_{x}^{u}$$\mu$$(\mu_{x}^{u})(\subset\overline{W^{u}(x)})$$\mu- almo\mathit{8}t$

Proof. Fix $0<\epsilon<1$ and let
be as in Lemma 2.3. By assumption
we can take and fix
such thatis not a point measure.
pointstwo
Choosedistinctand put $\tau=d(y_{1}, y_{2})/2(>0)$ .
Take arbitrarily $0<r<\tau/2$ and choose $\delta=\delta(r)>0$ such that
$F_{\epsilon}^{u}$$x_{0}\in \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\mu|F_{\epsilon}^{u})$$y_{1},$$\mu_{x_{0}}^{u}$$y_{2}\in \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\mu_{x0}^{u})$$(i=1,2)$ .

$\mu_{x_{0}}^{u}(U(yi, r))>\delta$(7)

Remark that$d(U(y_{1}, r),$ $U(y_{2}, r))= \inf\{d(x, y) :d(x, y_{1})<r, d(y, y_{2})<r\}>\tau$.

(8)

Since $U(y_{i}, r)(i=1,2)$ are open, by (7) there exists a large integer $M=$
$M(r)>0$ such that$\nu(U(yi, r))>\delta=\delta(r)$for$\nu\in \mathcal{M}(M)$with $\rho(\nu, \mu_{x0}^{u})<1/M$ . We can find$\rho(\mu_{x}^{u}, \mu_{x}^{u}0)<1/2M=1/2M(r)$(9)$(i=1,2)$$\epsilon’=\epsilon’(r)>0$

such that.

(10)$(k\geq n)\}$(11)$(x\in U(x_{0}, \epsilon)’\cap F_{\epsilon}^{u})$Note that depends on .
Let
be as in Lemma 2.1 and put$\epsilon’$$r$$\xi^{u}$$B_{m}^{u}(n)=\{x\in M|\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}(\rho(\mu xf^{k}\xi u,uf^{-}k+[k/2\mathrm{J}\xi^{u}\mu_{x})\leq 1/(f-[k/2]x))m,\leq 1/m$

forand, by Lemma 2.1(c) and Doob’s theorem. Thus there exists an increasing sequencesuchthat $\mu(B_{m}^{u}(n_{m}))\geq 1-1/2^{m+1}(m\geq 1)$ . Since $\mu(\cap^{\infty}k=mB_{k(}^{u}nk))\geq 1-1/2^{m}$ for$m\geq 1$ , we can find a Borel set
with $\mu(D_{m}^{u})\geq 1-2^{-m/2}$ satisfying$n,$ $m\geq 1.\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{n}B^{u}m(n)\subset B_{m}^{u}(n+1)$$\mu(\bigcup_{n=0^{B_{m}}}^{\infty}u(n))=1$$\{n_{m}\}$$D_{m}^{u}$$\mu_{x}^{u}(\mathrm{n}_{k=}^{\infty}B_{k}^{u}(mn_{k}))\geq 1-2^{-m/2}$$(x\in D_{m}^{u})$

.

(12)

Put
$K_{r}= \bigcap_{k=1}^{\infty}\bigcup_{m=k}^{\infty}(_{n}\bigcap_{=0}^{\infty}\bigcup_{\ell\geq n}f^{-}l(U(x_{0}, \epsilon’(r))\cap F_{\epsilon}u\cap Du)m)$$(0<r<\tau/2)$ .

Then $\mu(K_{r})=1(0<r<\tau/2)$ by ergodicity of . To obtain the conclusion it
sufficies to show thatis-chaotic set for $x\in K=\mathrm{n}n\geq 1K1/n$ .
$\mu$$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\mu_{x}^{u})$$\mathrm{a}*$23

To do this fix $x\in K_{r}(r=1/n, n\geq 1)$ and suppose that nonempty opensets
andsatisfy$U_{1}$$U_{2}$$U_{1}\cap U_{2}\neq\emptyset$,$(j=1,2)$ .

$U_{j}\cap \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\mu_{x}^{u})\neq\emptyset$

Choose $m_{0}>0$ with
$0<2^{-m\mathrm{o}/2}< \min\{\mu^{u}x(Uj) :j=1,2\}$

and $m_{0}\geq 2M$ .

Since $x\in K_{r}$ , by the definition of , there exist $m_{1}>m_{0}$ and a sequence of
positive integers
withsuch that$K_{r}$$\ell_{k}>n_{k}$$\{\ell_{k}\}_{k}$$f^{l_{k}}(x)\in U(x_{0}, \epsilon’(r))\cap F_{\epsilon m}^{u}\mathrm{n}D^{u}1$$(k\geq 1)$

.

(13)

Since
$\mu_{x}^{u}(f^{-l}k(B^{u}k(nk)))$$\geq\mu_{x}^{u}(f^{-f}k(\bigcap_{kk}^{\infty}Bu(=m1nk)))$$=\mu_{f^{l_{k}}(}^{u}x)(\cap k=m_{1}B_{k(}^{u}\infty nk))$$\geq 1-2^{-m1/2}\geq 1-2^{-m/2}0$$(k\geq m_{1})$$\mu_{x}^{u}(U_{j}\cap f^{-}\ell_{k}(B_{k}u(nk)))\geq\mu_{x}^{u}(U_{j})-2-m_{0}/2>0$$(k\geq m_{1})$by (12), we haveThen, by the definition of$\{\mu_{x}^{u}|x\in M\}$.

, we can choose$z_{j}=z_{j}(k)\in U_{j}\cap f^{-}\ell kB_{k}^{u}(n_{k})$with$\eta^{u}(x)=\eta^{u}(z_{j})$$\eta^{u}(f^{l_{k}}(z_{j}))$we haveand sofor $j=1,2$ and $k\geq m_{1}$ . Thus we have
. Since$\mu_{f^{f}k}^{u}(x)\mu^{u}f^{l}k(zj)=$$\eta^{u}(f^{\ell_{k}}(x))=$$f^{\ell_{k}}(z_{j})\in B_{k}^{u}(n_{k})\subset B_{k}^{u}(\ell_{k})$, by (11), (14)$\rho(\mu_{f^{t_{k(z_{j}}}}^{f^{t}}, \mu ft_{k}(x)))=\rho(k\xi uuftk\xi^{u},f\mu t_{k})\mu f(\mathcal{Z}_{j}(t_{k}z_{j}))u\leq 1/k\leq 1/m_{0}\leq 1/2M$

diam $(f^{-f}k+[\ell k./2]\xi u(ft_{k}-[t_{k}/2](_{Z}j)))\leq 1/k$

(15)

for $j=1,2$ and $k\geq m_{1}$ . Thus, by (9), (10), (13) and (14),$\mu_{z_{j}}^{\xi^{u}}(f^{-}l_{k}U(y_{i}, r))=\mu^{f^{t_{k\xi}u}}t_{k}(zj)(fU(yi, r))>\delta$Since$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\mu_{z_{\mathrm{j}}}\xi^{u})\subset\xi^{u}(z_{j})$.

, we have$\xi^{u}(z_{j})\cap f-fkU(y_{i}, r)\neq\emptyset$

for $1\leq i,j\leq 2$ and $k\geq m_{1}$ . For$k$large enough, by (15) we may assume$z_{j}\in f^{-}l_{k}+[^{\ell_{k}}/2]\xi^{u}(f\ell_{k}-[\ell_{k}/2](Z_{j}))\subset U_{j}$.

Therefore$U_{j}\cap f^{-\ell_{k}}U(y_{i}, r)$$\supset f^{-\ell_{k}[^{\ell_{k}/2]}}+\xi u(f\ell k-[\ell_{k}/2](Zj))\cap f^{-}l_{k}U(y_{i}, r)$$\supset\xi^{u}(_{Z_{j}})\cap f-\ell_{k}U(y_{i}, r)\neq\emptyset$

24

for $1\leq i,j\leq 2$ and$k$large enough. Take$b_{i,j}=b_{ii}(k)\in f^{-\ell_{k}}(U(yi, r))\cap Uj$

for $1\leq i,j\leq 2$ . Then we have that for $1\leq i,$ $j\leq 2$
and $d(f^{l_{k}}(b1,1),$ $fl_{k}(b_{1,2}))<r=1/n$,$d(f^{\ell_{k}}(b_{1,1}), f^{\ell}k(b_{2,2}))>\tau$$b_{i,j}\in U_{j}$by (8). This implies that$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\mu_{x}^{u})$Proof of Theorem

5is$\mathrm{a}*$-chaotic set for$x\in K=\mathrm{n}n\geq 1K1/n\cdot\square$$\mathrm{C}$The purpose of this section is to show Theorem C. Let be a -diffeomorphism-manifold $M$ and let be an ergodic SRB measure. As deof a closed. Thus we
formula holds: i.e.
scribed in \S 1 thein$h_{\mu}(f)\geq\max\{\lambda_{i}\}>0$(A)
mentioned
because satisfies the conditionhave
is -chaotic set for -almost all $x\in M$ .
\S 1. Therefore, by Theorem-chaotic set we need the following lemma.
isTo show that$C^{2}$$f$$C^{\infty}$$\mu$$\mathrm{P}\mathrm{e}\sin$)$h_{\mu}(f)=\Sigma_{\lambda_{i}>0^{\lambda_{i}m}i}$$\mathrm{s}$$\mu$$\mathrm{B},$$\overline{W^{u}(X)}$$\overline{W^{S}(x)}$$\mathrm{a}*$$\mu$$\mathrm{a}*$Lemma 5.1 ([8], Corollary 6.1.4) Let be an ergodic measure satisfying
withbe as in Lemma 2.1 and let be the density ofPesin’s formula, let
is a strictly positive function on. Then at -almost all
respect to-leaves.
Lipschitz along
andbe as in \S 2. Then, by Proposition 2,
andLet-almost all $x\in M$ . Therefore, to obtain theforis a -chaotic set
conclusion it sufficies to show the following.
$\mu$$m_{x}^{u}$$\xi^{u}(x)$$x_{f}\psi$$\mu$$W^{u}$$\log\psi i\mathit{8}$$\{\mu_{x}^{u}|x\in M\}$$\eta^{u}$$\overline{W^{u}(x)})$$*$$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\mu_{x}^{u})(\subset$$\mu$

Proposition 3 If
all $x\in M$ .

$\mu$

is an $sRBmea\mathit{8}ure$ , then supp

Proof. We first show thatis finer than , for -almost all $z\in M$$\xi^{u}(x)\subset \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\mu^{u}x)$$\xi^{u}$$\mu_{x}^{\xi^{u}}$$\psi$$\xi^{u}$$\eta^{u}$$(\mu_{x}^{u})=\overline{W^{u}(x)}$for -almost$\mu$

for -almost all $x\in M$ . Since
$\mu$

$\mu$

$\int\mu_{x}^{\xi^{u}}(\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\mu^{u}z))d\mu z(ux)=\mu^{u}z(\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\mu_{z}^{u}))=1$

for
Then
Lemma 5.1 we have that$\mu_{x}^{\xi^{u}}(\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{P}(\mu_{z})u)=1$$\mu_{z}^{u}$-almost all$x$. Since.

$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\mu^{u}z)$

is closed, by$\xi^{u}(x)\subset \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\mu_{z})u\sup=\mathrm{P}(\mu^{u}x)$

for -almost all .
-almost all . Therefore
for -almost all , we haveSince-almost all . By Lemma 2.1 (d)for$f_{*}\mu_{x}^{u}=\mu_{f^{x}}^{u}$$\mu$$\xi^{u}(x)\subset \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\mu_{x}^{u})$$x$$\mu_{z}^{u}$

$x$

$\mu$

$x$

$\mu$

$f(\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\mu_{x}u))=\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\mu_{fx}^{u})$

for

$x$

$W^{u}(x)$$= \bigcup_{n=0}^{\infty}fn\xi^{u}(f-n(x))$$\subset\bigcup_{n=0}^{\infty}fn(\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\mu^{u}f-n(x)))$$\subset \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\mu_{x}^{u})$for -almost all $x\in M$ . Therefore$\mu$$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\mu^{u}x)=\overline{W^{u}(x)}$for -almost all
$\mu$

.

$x\in M\square$

25

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