数理解析研究所講究録 1220 巻 2001 年 54-62

54

Local diffeomorphisms with positive entropy
and chaos in the sense of Li-Yorke
Naoya Sumi(鷲見直哉)

Abstract-local diffeomorphism with positive entropy on anWe show that if is adimensional closed manifold $(n\geq 2)$ then is chaotic in the sense of Li-Yorke.
$C^{2}$$f$$f$

1Introduction
We study chaotic properties of dynamical systems with positive entropy. Notions of chaos
have been given by Li and Yorke [15], Devaney [5] and others. It is well known that if acontinuous map of an interval has positive entropy, then the map is chaotic according to
the definition of Li and Yorke (cf. [2]). For invertible maps the following holds: let be-manifold. If the topological entropy of is positive,
a -diffeomorphism of aclosed
then is chaotic in the sense of Li-Yorke [31].
In this paper we show the following:$f$$C^{2}$$f$$C^{\infty}$$f$

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

From this theorem we obtain the following corollary.
-local diffeomorphism of a closed
Corollary
Let be ainvertible, then is chaotic in the sense of Li- Yorke.
$f$$\mathrm{B}$$C^{2}$$C^{\infty}$-manifold. If$f$is not$f$

First we shall explain here the definitions and notations used above. Let $X$ be acompact
metric space with metric and let : $Xarrow X$ be acontinuous map. Asubset of $X$ is ascrambled set of if there is apositive number $\tau>0$ such that for any , $y\in S$ with $x\neq y$ ,
$d$

$S$

$f$

$x$

$f$

1. $\lim\sup_{narrow\infty}d(f^{n}(x), f^{n}(y))>\tau$ ,2. $\lim\inf_{narrow\infty}d(f^{n}(x), f^{n}(y))=0$ .
If there is an uncountable scrambled set of , then we say that is chaotic in the senseof Yorke. Li and Yorke showed in [15] that if $f:[0,1]arrow[0,1]$ is acontinuous map withaperiodic point of period 3, then is chaotic in this sense. Note that any scrambled set
contains at most one point which does not satisfy the following: for any periodic point$p\in X$ ,
$S$

$f$

$f$

$L\mathrm{i}$

$f$

$x$

$\lim_{narrow}\sup_{\infty}d(f^{n}(x), f^{n}(p))>0$.

For another sufficient condition for the chaos in the sense of Li-Yorke, the readers may refer
to [4], [7], [8], [9], [10], [11], [19], [20], [34].
Concerning the chaos in the sense of Li-Yorke, Kato introduced the notion of
as follows: let $F$ be aclosed subset of $X$ . Amap $f:Xarrow X$ i\S *-chaotic on $F$ (in the senseof Li-Yorke) if the following conditions are satisfied:$”*\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{o}\mathrm{s}$1. there is $\tau>0$ with the property that for any nonempty open subsets and of withand for any natural number $N$ , there is $n\geq N$ such that $d(f^{n}(x), f^{n}(y))>\tau$for some $x\in U$ , $y\in V$ , and$U$$U\cap V=\emptyset$

$V$

$F$

55
2. for any nonempty open subsets , of and any $>0$ there is anatural number$n\geq 0$ such that $d(f^{n}(x), f^{n}(y))<\epsilon$ for some $x\in U$ , $y\in V$ .
$U$

$V$

$F$

$\epsilon$, isSuch aset is called-chaotic set. If is ascrambled set, then the closure of-chaotic set. In [10] Kato showed that the converse is true. This is stated precisely as
follows:$F$$S$$\mathrm{a}*$$S,\overline{S}$$\mathrm{a}*$Lemma 1([10], Theorem 2.4) Let $X$ be a compact metric space and let $F$ be a closed
subset of X. If : $Xarrow X$ is continuous and is -chaotic on $F$ , then there is anset$S\subset F$ such that
. If $F$ is perfect ( . $F$ has no isolated
is a scrambled set of andpoints), we can choose as a countable union of Cantor sets.
$f$

$*$

$S$

$F_{\sigma}$$\overline{S}=F$$f$$|..e$$S$

By this lemma, to show the existence of uncountable scrambled sets it suffices to show
the existence of-chaotic set.
To obtain Theorem Awe consider the inverse limit system of . Let $M$ be aclosed-manifold and let be the distance for $M$ induced by aRiemannian metric
on $TM$ .
Letdenote the product topological space $M^{\mathrm{Z}}=\{(x:) : x:\in M,i\in \mathbb{Z}\}$ . Thenisbycompact. We define acompatible metric for$\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{e}\mathrm{c}\mathrm{t}*$

$f$

$d$

$C^{\infty}$$||\cdot||$$M^{\mathrm{Z}}$$M^{\mathrm{Z}}$$\tilde{d}$$M^{\mathrm{Z}}$.

$\tilde{d}((x\dot{.}), (y:))=\sum_{:=-\infty}^{\infty}\frac{d(x_{\dot{1}},y\dot{.})}{2^{|\cdot|}}$$((x_{i}), (y:)\in M^{\mathrm{Z}})$.

For $f:Marrow M$ acontinuous surjection, we let{$M_{f}=$

Then
is closed subset ofby . Ahomeomorphism$M_{f}$: $X:\in M$ and$(\mathrm{x}\{)$$M^{\mathrm{Z}}$. The space$\tilde{f}:M_{f}arrow M_{f}$$f$$\{(\mathrm{x}_{\mathrm{j}})=x:+1,i\in \mathbb{Z}\}$.

is called the inverse limit space constructed, which is defined by$M_{f}$$\tilde{f}((x:))=(f(x:))$for all,$(\mathrm{x}\{)\in M_{f}$: $M_{f}arrow M$ the projection definedis called the shift map determined by . We denote as. Thenbyholds. Remark that is chaotic in the sense of Li-Yorke
if and only if so is .
We can show that the topological entropy, $h(f)$ , of -f coincides with that of . Indeed, for
an -invariant probability measure , we can find an -invariant probability measure such). Let usthat $\nu(A)=P_{*}^{0}\mu(A)(=\mu((P^{0})^{-1}A))$ for any Borel set $A\subset M$ ([18] Lemma
respectively. Thenandthe metric entropy of $(M, f, \nu)$ anddenote as([25] Lemma 5.2). Therefore, the conclusion is obtained
we haveby the variational principle ([32] Theorem 8.6).
We say that adifferentiate map $f:Marrow M$ is alocal diffeomorphism if for $x\in M$ there
is an open neighborhoodof in $M$ such that $f(U_{x})$ is open in $M$ and: $U_{x}arrow f(U_{x})$is adiffeomorphism. Since $M$ is connected, then the cardinal number of $f^{-1}(x)$ is constant.
This constant is called the covering degree of . If the covering degree of is greater thanone, $(M_{f}, M, C, P^{0})$ is fiber bundle where $C$ denotes the Cantor set (see [1] Theorem 6.5.1).
Let be aBorel probability measure on
and let be the Borel a-algebra oncompleted with respect to . For ameasurable partition ofand
we denote as
the element of the partition 4which contains the point . Then there exists afamilyof Borel probability measures satisfying the following conditions:$P^{0}$$f$$P^{0}\circ\tilde{f}=f\circ P^{0}$$(x_{i})\vdasharrow x_{0}$$f$$\tilde{f}$$\tilde{f}$

$f$

$f$

$\nu$

$\mu$

$\mathrm{I}\mathrm{V}8.3$$h_{\nu}(f)$$(M_{f},\tilde{f},\mu)$$h_{\mu}(\tilde{f})$$h_{\nu}(f)=h_{P^{\mathrm{O}}\mu}.(f)=h_{\mu}(\tilde{f})$$U_{x}$$x$$f|_{U_{*}}$

$f$

$f$

$M_{f}$

$\mu$

$\mu$

$B$

$M_{f}$$M_{f}$$\xi$$\tilde{x}\in M_{f}$$\tilde{x}$$\xi(\tilde{x})$$\{\mu\frac{\epsilon}{x}|\tilde{x}\in M_{f}\}$

1. for2.

$\tilde{x},\tilde{y}\in M_{f}$$\mu\frac{\epsilon}{x}(\xi(\tilde{x}))=1$

if$\xi(\tilde{x})=\xi(\tilde{y})$thenfor -almost all$\mu$

3. for $A\in B$ afunction$\mu\frac{\epsilon}{x}=\mu\frac{\epsilon}{y}$$\tilde{x}\in M_{f}$$\tilde{x}\vdash*\mu\frac{\epsilon}{x}(A)$,,is measurable and$\mathrm{f}\mathrm{i}(\mathrm{A})=\int_{M_{f}}\mu\frac{\epsilon}{x}(A)d\mu(\tilde{x})$.

The family
is called acanonical system of conditional measures for(see [26] for more details).
To prove Theorem Ait suffices to show the following theorem$\{\mu\frac{\epsilon}{x}|\tilde{x}\in M_{f}\}$

$\mu$

and

$\xi$

56
Theorem
Let be a-local diffeomorphism of a closed-manifold $M$ and let be.
an -invariant ergodic Borel probability measure on
If the metric entropy of is positive, then there exists a measurable partition ofsuch thatis a $perfect*$ -chaotic set for -almost all.
$C^{2}$$f$$\mathrm{C}$$C^{\infty}$$\mu$$\tilde{f}$$M_{f}$$\mu$$M_{f}$$\eta$$supp( \mu\frac{\eta}{x})$$\tilde{x}\in M_{f}$$\mu$Here the supportof afinite measure is the smallest closed set with. Equivalently,
is the set of allwith the property thatfor
any open containing .
Let us see how Theorem Afollows from Theorem C. We know that:
where
is the set of all -invariant ergodic probability measures (cf.[27]).
Thus, if $h(\tilde{f})=h(f)>0$ , then we can choose
with. Therefore, byTheorem and Lemma 1, is chaotic in the sense of Li-Yorke.
$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\nu)$$\nu(M_{f})$

$U$

$C$

$\nu$

$\mathrm{v}(\mathrm{C})=$$\mathrm{v}(\mathrm{U})>0$$\tilde{x}\in M_{f}$$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\nu)$$\tilde{x}$$h( \tilde{f})=\sup\{h_{\mu}(\tilde{f})$$\mu\in \mathcal{M}_{e}(\tilde{f})\}$$\mathcal{M}_{e}(\tilde{f})$$\tilde{f}$$h_{\mu}(\tilde{f})>0$$\mu\in \mathcal{M}e(f)$$f$$\mathrm{C}$2Key Lemmas
In this section we prepare some lemmas which need to prove Theorem C. Let be a
local diffeomorphism of aclosed-manifold $M$ and be an -invariant ergodic Borel. As in the previous section we denote asprobability measure on
withcompleted with respect to . For -almost all $\tilde{x}=(x_{i})\in M_{f}$ ,the Borel -algebra on
there exist asplitting of the tangent spaceand real numbers
such that$C^{2_{-}}$$f$$C^{\infty}$$\tilde{f}$$\mu$$h_{\mu}(\tilde{f})>0$$M_{f}$$B$$M_{f}$$\sigma$

$\mu$

$\mu$

$T_{x_{0}}M=\oplus_{=1}^{\epsilon(x_{0})}.\cdot E.\cdot(\tilde{x})$$\lambda_{1}(x_{0})<$$\mathrm{s}(\mathrm{x}\mathrm{o})<\cdots<\lambda_{\iota(x_{0})}(x_{0})$(a) the maps

andand $s(x_{0})$ are measurable, moreover, $s(x_{0})$ are -invariant $(i=1, \cdots, s(x_{0}))$ ,$\tilde{x}\vdash\rangle$$\lambda_{:}(x_{0})$$E.\cdot(\tilde{x})$,$\lambda_{:}(x_{0})$$E_{\dot{1}}(\tilde{f}(\tilde{x}))=\mathrm{D}\mathrm{X}\mathrm{o}\mathrm{f}(\mathrm{E}\mathrm{i}(\mathrm{x}))$

$f$

(b)

$\lim_{narrow\pm\infty}\frac{1}{n}\log||(.D_{x_{0}}f^{|n|})^{\pm 1}(v)||=\lambda:(x_{0})$(c)$n arrow\lim_{\pm\infty}\frac{1}{n}\log|\det(D_{xx_{0}}f^{|n|})^{\pm 1}|=.\sum_{\dot{|}=1}^{(xx_{0})}\lambda:(x_{0})\dim E:(\tilde{x})$($0\neq v\in \mathrm{E}\mathrm{i}(\mathrm{x}),$$i=1$ ,$\cdots$,

and$\mathrm{s}(\mathrm{x}\mathrm{o})$, ,([21], [33], [29], [30]). The numbers
are called Lyapunov exponents,
of at . Since is ergodic, we can put
and $m:=\dim E.(\tilde{x})$$(i=1, \cdots, s)$ for-almost all $\tilde{x}=(x:)\in M_{f}$ .
Awell-known theorem of Margulis and Ruelle [28] says that entropy is always bounded. Since has
above by the sum of positive Lyapunov exponents; i.e.
positive entropy, we have $0<h_{\mu}( \tilde{f})=h_{P^{0}\mu}.(f)\leq\max\{\lambda:\}=\lambda.$.Fix $0< \lambda<\min\{\lambda$ :. From [24], [29] and [30] there are measurable functions
andwiththe following properties: For $\tilde{x}=(x:)\in M$ we put$\lambda_{1}(x_{0})$$f$$x_{0}$$\cdots$$\lambda_{e(x_{0})}(x_{0})$$\lambda_{:}=\mathrm{s}(\mathrm{x}\mathrm{o})$$s=\mathrm{s}(\mathrm{x}\mathrm{o})$

$\mu$

$\mu$

$\tilde{f}$$h_{P^{0}\mu}.(f) \leq\sum_{\lambda_{i}>0:}\lambda m:$,$\lambda_{:}>0\}$$\tilde{W}_{lo\mathrm{c}}^{u}(\tilde{x})=\{\tilde{y}=(y.\cdot)\in M_{f} :$\tilde{\beta}>\tilde{\alpha}>0$.

$\tilde{\gamma}>1$d(x_{0},y\mathrm{o})\leq\tilde{\alpha}(\tilde{x}), d(X:,y:)\leq\tilde{\beta}(\tilde{x})e^{-:\lambda} (:\geq 1)\}$.

Then(a) the map$P^{0}$

restricted tois a$\tilde{W}_{lo\mathrm{c}}^{u}(\tilde{x})$is injective,-submanifold of the ball(b)$P^{0}(\tilde{W}_{loc}^{u}(\tilde{x}))$(c)$T_{x_{0}}P^{0}(\tilde{W}_{lo\mathrm{c}}^{u}(\tilde{x}))=\oplus_{\lambda_{:}>0}E_{\dot{1}}(\tilde{x})(\neq\{0\})$(d) $d(y:,z:)$$C^{2}$$\leq\tilde{\gamma}(\tilde{x})d(y_{0},z_{0})e^{-:\lambda}$for$(y_{n})$,$\{y\in M : \mathrm{d}(\mathrm{x}\mathrm{o},\mathrm{y})\leq\tilde{\alpha}(\tilde{x})\}$

for -almost all$\mu$$(\mathrm{z}\mathrm{n})\in\tilde{W}_{lo\mathrm{c}}^{u}(\tilde{x})$$\tilde{x}\in M_{f}$

,

,

.

For the case when is invertible we may refer to [6], [22] and [23].
. Put
Let and be measurable partitions of
for
.
then
for
means that for -almost all
.
has
$f$

$\xi$

$M_{f}$$\eta$$(\tilde{f}^{n}\xi)(\tilde{x})=\tilde{f}^{n}(\xi(\tilde{f}^{-n}(\tilde{x})))$$\tilde{x}\in M_{f}$$\eta\leq\xi$$\tilde{f}^{n}\xi=\{\tilde{f}^{n}C : C\in\xi\}$$\mu$$n\in \mathbb{Z}$$\tilde{x}\in M_{f}$

and

one

$\xi(\tilde{x})\subset\eta(\tilde{x})$Lemma 2Let

$f$

and

$\mu$

be as above. Then there eists a measurable partition 4of$M_{f}$

such

57
(a)

$\xi\leq\tilde{f}^{-1}\xi$

,
all

(b)

$\tilde{x}\in M_{f}$,$\xi(\tilde{x})\subset\tilde{W}_{loc}^{u}(\tilde{x})$andcontains a neighborhood of$\xi(\tilde{x})$$for\mu- almostin\tilde{W}_{loc}^{u}(\tilde{x})$(c)$\bigvee_{n=0}^{\infty}\tilde{f}^{-n}\xi$$\tilde{x}$openis the partition into points.

This lemma is similar to [13] Proposition 3.1, [16] Proposition 5.2 and [17] Lemma 2.2.
So we omit the proof.
Let denote the family of all nonempty closed subsets ofand define ametricby$\mathrm{C}$$d_{H}$$M_{f}$$(A, B\subset(?)$$d_{H}(A, B)= \max\{\sup_{b\in B}d(A, b),\sup_{a\in A}d(a, B)\}$

where $d(A, b)= \inf\{d(a, b) : a\in A\}$ . Then it is known thatis acompact metricspace (cf.[12]). If is ameasurable partition, thenis measurable. Indeed,this follows from [3] Theorems III.2, III.9, III.22 and the fact thatis aBorel subset of. For $A\subset M_{f}$ we put diam(A). Thenwe have diam(A)Since
is measurable, }is also ameasurable function. By Lemma 2(c) we have that for almost all$(\mathrm{C}, d_{H})$$\tilde{x}\succ*\overline{\xi(\tilde{x})}\in \mathrm{C}$$\xi$$\{(\tilde{x}, \xi(\tilde{x})):\tilde{x}\in M_{f}\}$$M_{f}\cross M_{f}$$= \sup\{d(\tilde{x},\tilde{y}) :$\tilde{x}\vdasharrow\overline{\xi(\tilde{x})}\in \mathrm{C}$$=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}(\mathrm{A})$$\tilde{x}$\mathrm{x},\mathrm{y}\in A\}$

$arrow \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}(\xi(\tilde{x}))$$\tilde{x}\in M_{f}$

$\mu$

(1)

$\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}((\tilde{f}^{-n}\xi)(\tilde{x}))arrow 0$

as.
Let 4and be measurable partitions ofand letbe acanonical systemof conditional measures for and . The mean conditional entropy of with respect to 4is
defined by$narrow\infty$$\{\mu\frac{\epsilon}{x}|\tilde{x}\in M_{f}\}$$M_{f}$$\eta$

$\xi$

$\mu$

$\eta$$H_{\mu}( \eta|\xi)=\int-\log\mu\frac{\epsilon}{x}(\eta(\tilde{x}))d\mu(\tilde{x})$(see [27] for details).

Lemma 3Let

$f$

and

$\mu$

be as above and let 4be as in Lemma 2. Then,$h_{\mu}(\tilde{f})=H_{\mu}(\overline{f}^{-1}\xi|\xi)$.

For the case when is invertible this lemma is proved by Ledrappier and Young [14].
We recall that if the covering degree of is greater than one, then (, Af, , ) is afiber
bundle where $C$ denotes the Cantor set. In view of this fact, the above lemma can be provedby almost the same arguments as the proof of [14] Corollary 5.3 and [16] Corollary 7.1 withsome slight modifications. Here we omit the proof.
By Lemma
we have that
. Let us introduce ameasurable
partition defined by. Then we have
. For simplicity put
$f$

$f$

$C$

$M_{f}$$P^{0}$$\xi\geq\tilde{f}\xi\geq\tilde{f}^{2}\xi\geq\cdots$$2(\mathrm{a})$$\tilde{f}\eta=\eta$$\eta=\bigwedge_{i=0}^{\infty}\tilde{f}^{i}\xi$pi$= \mu\frac{\eta}{x}$and$\mu_{\overline{x}}^{n}=\mu\frac{\overline{f}}{x}n\epsilon$$(n\in \mathbb{Z})$By Doob’s theorem it follows that for a -integrable function$\mu$$\psi$.
:$M_{f}arrow \mathbb{R}$(2)$\int\psi d\mu_{\overline{x}}=\lim_{narrow\infty}\int\psi d\mu_{\overline{x}}^{n}$, by the uniqueness of acanonical system offor almost all . Since
andconditional measures (cf.[26]) we have that for almost all$\tilde{f}\eta=\eta$$\tilde{x}$$\mu$$\tilde{f}_{*}\mu=\mu$$\tilde{x}$$\mu$$\overline{f}_{*}\mu_{\overline{x}}=\mu_{\overline{f}\overline{x}}$and$\tilde{f}_{*}\mu_{\overline{x}}^{n}=\mu_{\overline{f}\overline{x}}^{n+1}$$(n\in \mathbb{Z})$

.

(3)

Here
for aBorel probability measure on
and $A\in B$ .
$C(M_{f})$Letbe the Banach space of continuous real-valued functions ofwith the $\sup$, and letnorm
be aset of all Borel probability measures onwith the weak$(\tilde{f}_{*}\nu)(A)=\nu(\tilde{f}^{-1}A)$$\nu$$M_{f}$$M_{f}$$|\cdot|_{\infty}$$\mathcal{M}(M_{f})$$M_{f}$

58.} which is dense

topology. Since $C(M_{f})$ is separable, there exists acountable set. For v, E $M(M’)$ definein$\{\mathrm{p}_{\mathrm{i}_{\mathrm{t}}}\mathrm{p}_{2\mathrm{t}}\ovalbox{\tt\small REJECT}$$v’$$C\ovalbox{\tt\small REJECT} M_{f}\ovalbox{\tt\small REJECT}$$\rho(\nu, \nu’)=\sum_{n=1}^{\infty}\frac{|\int\varphi_{n}d\nu-\int\varphi_{n}d\nu’|}{2^{n}|\varphi_{n}|_{\infty}}$

Then is acompatible metric for, we haveholds for$\mathrm{M}$$\rho$(Mf) and$(\mathcal{M}(M_{f}), \rho)$.

is compact (cf.[18]). Since (2)$\{\varphi:\}$(4)$\mu_{\overline{x}}=\lim_{narrow\infty}\mu_{\overline{x}}^{n}$for -almost all . For $\nu\in \mathcal{M}(M,)$ and ameasurable partition , by the definition of, the map
conditional measures$\tilde{x}$

$\xi$

$\mu$

$\{\nu\frac{\epsilon}{x}\}$$M,$

is measurable for $n\geq 1$ and thus

Lemma 4Let ,
with
$f$

$\mu$

$\ni\tilde{x}\vdasharrow\int\varphi_{n}d\nu\frac{\epsilon}{x}$$\tilde{x}\vdasharrow\nu\frac{\xi}{xx}\in \mathcal{M}(M_{f})$is measurable.

be as above. Then forandsatisfying the map$\{\mu_{\overline{x}}|\overline{x}\in M_{f}\}$$\epsilon$$>0$ there eists a closed set$\mu(F_{\epsilon})\geq 1-\epsilon$$F_{\epsilon}$$F_{\epsilon}\ni\tilde{x}\vdash\rangle\mu_{\overline{x}}\in \mathcal{M}(M_{f})$is continuous.

be as above and let $>0$ . Since
Let. Proof.
with, by Lusin’s theorem there exists aclosed set$\{\varphi_{1}, \varphi_{2}, \cdots\}$$|$$F_{\dot{1}}$$F_{\epsilon}= \bigcap_{=1}^{\infty}.\cdot$$(: \geq 1)$

is measurable forsatisfying$\mu(F_{\dot{1}})\geq 1-\epsilon/2^{:}$: continuous.

$F. \cdot\ni\tilde{x}\vdasharrow\int\varphi:d\mu_{\overline{x}}$Then$\overline{x}\vdash\nu\int\varphi:d\nu\frac{\epsilon}{x}$$\epsilon$$\geq 1$

F. has the desired property.
$\cdot$cldenote the restriction of to $E$ , i.e. $\nu|E(A)=\nu(A\cap E)$
and $E\in B$ let
the closed andand. Clearly
is afinite measure. We denote as
for
$>0$
be as
respectively.
Let
and radius
open balls inwith center
we can find such that
. For
and
above and let

For$\nu|_{E}$$\nu\in \mathrm{M}(\mathrm{M}\mathrm{f})$$A\in B$$\nu$

$U(\overline{x},r)$$B(\overline{x},r)$$\nu|E$$r$$\tilde{x}\in M_{f}$$M_{f}$$\nu\in \mathcal{M}(Mf)$$\{\varphi_{1},\varphi_{2}, \cdots\}$$\epsilon>0$$\tilde{x}\in \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\nu)$$\int_{U(\overline{x},\epsilon)}\varphi:d\nu>\int\varphi:d\nu-\epsilon$

Since the inequality holds for$i$.

sufficiently close to , we can easily prove that$\nu’$$\nu$$\mathcal{M}(M_{f})\ni\nu\vdasharrow \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\nu)\in \mathrm{C}$is lower semi-continuous and so the map is measurable ([3] Corollary III.3). Sincediam(supp(i/)) is lower semi-continuous,$\mathrm{M}(\mathrm{M}\mathrm{f})=$$=$is aclosed set of$\mathrm{M}$(Mf). Since{{$\nu\in \mathrm{M}(\mathrm{M}\mathrm{f})$$\nu\in \mathrm{M}$:(Mf) :$\nu$

is apoint measure}$\dim(\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\nu))=0$$(\tilde{f}^{n}\xi)(\tilde{x})\subset\eta(\overline{x})$$\mu$$\tilde{x}\in M_{f}$$\mu$$(n\in \mathrm{Z})$.

Lemma 5Let , andhas no isolated points$f$}, we have$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\mu_{\overline{x}}^{n})\subset \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\mu_{\tilde{x}})$for -almost all$\nu\vdash\star$$\{\mu_{\overline{x}}|\tilde{x}\in M_{f}\}$be as above. Then for -almost all$\mu$$\tilde{x}\in M$,$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\mu_{\overline{x}})$59
Proof. Let

and

$\xi$

$P_{n}=\{\tilde{x}\in M_{f} :$\mu_{\overline{x}}^{n}$be as above. Then it is easily checked that for\mu_{\overline{x}}^{n}\in \mathcal{P}(Mf)\}\supset${:$\overline{x}\in M_{f}$$\mu_{\overline{x}}|_{(\overline{f}^{\mathrm{n}}\xi)(\overline{x})}$$n\in \mathrm{Z}$is apoint measure}.

If this lemma is false, then there exists ameasurable set with positive measure such thatfor any belonging to the set,
has an isolated point. Since$(karrow\infty)$ by (1), we have $\mu(P_{-k})>0$ forlarge enough. Put
andthen $\mu(P)=1$ because is ergodic.
By (3) we have$\overline{x}$$\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}((f^{-k}\xi)(\tilde{x}))arrow 0$$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\mu_{\overline{x}})$$k$$P= \bigcap_{j\geq 1}\bigcup_{n\geq \mathrm{j}}\tilde{f}^{n}P_{-k}$$\mu$$\tilde{f}^{n}(P_{-k})$$=\{\tilde{f}^{n}(\tilde{x})\in M_{f} :$=\{\tilde{x}\in M_{f} :\tilde{f}_{*}^{n}\mu_{\overline{f}^{-n}\overline{x}}^{-k}\in \mathcal{P}(M)\}$$=\{\overline{x}\in M_{f} :\mu_{\overline{x}}^{n-k}\in \mathcal{P}(M_{f})\}$$=P_{n-*}$and sosuch that
have
Since

$P= \bigcap_{\mathrm{j}>1}\bigcup_{n\geq j}Pn-k$$\tilde{x}\in P_{n}$$\mu_{\overline{x}}\in \mathcal{P}(M_{f})$$\xi\geq \mathrm{y}7$:for$i\geq 0$. Thus, for. Since\mu_{\overline{xx}}^{-k}\in \mathcal{P}(M)\}$$(n\in \mathbb{Z})$

,

.

.

there exists an increasing sequence(by (4)) andfor , we$\tilde{x}\in P$$\mu_{\overline{x}}=\lim_{:arrow\infty}\mu_{\overline{x}}^{n}$

.
for
and
is apoint measure forfor -almost all . Therefore$\mu_{\overline{x}}^{n}$$\{n:\}:\geq 0$$\in \mathcal{P}(M_{f})$$i$$\tilde{x}\in P$$\mu_{x}$$\mu\frac{\epsilon}{x}((\overline{f}^{-1}\xi)(\tilde{x}))=1$$\mu$-almost all$\tilde{x}\in M_{f}$, so is$\mu\frac{\epsilon}{x}$. Thus$\tilde{x}$$\mu$$\bullet$$h_{\mu}( \overline{f})=H_{\mu}(\tilde{f}^{-1}\xi|\xi)=\int-\log\mu\frac{\epsilon}{x}((\overline{f}^{-1}\xi)(\tilde{x}))d\mu(\tilde{x})=0$

by Lemma 3. This is acontradiction.
$\square$

3Proof of Theorem$\mathrm{C}$In this section we will prove Theorem C. Let
Lemma 5,is perfect for -almost all
it suffices to show the following.
$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(/\mathrm{i}\mathrm{i})$

$\mu$

$f$

,

$\mu$

,

and
be as in \S 2. By. Therefore, to obtain the conclusion$\eta$$\overline{x}\in M_{f}$$\{\mu_{\overline{x}}|\tilde{x}\in M_{f}\}$

Proposition 1If
is not a point measure for -almost all-chaotic set for -almost all.
$\mu$$\mu_{\overline{x}}$$*$$\tilde{x}\in M_{f}$, then$supp(\mu_{\overline{x}})$

is

$a$

$\overline{x}\in M_{f}$$\mu$Proof. The proof of this proposition is similar to that of [31] Proposition 2. However,for completeness we give the proof.
Fix $0<\epsilon$ $<1$ and let
be as in Lemma 4. By assumption we can take and fix
such thatis not apoint measure. Choose two distinct points
and put. For $0<r<\tau/2$ we can take $=6(\mathrm{r})>0$ with
$F_{\epsilon}$$\tilde{x}_{0}\in \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\mu|F_{\epsilon})$$\tilde{y}_{1},\overline{y}_{2}\in$$\mu_{\overline{x}_{0}}$$\tau=d(\tilde{y}_{1},\overline{y}_{2})/2(>0)$$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\mu_{\overline{x}_{0}})$$\delta$$\mu_{\overline{x}_{0}}(U(\tilde{y}.\cdot,r))>\delta$

Since$U(\overline{y}:,r)$$1/m’$($\nu\in \mathrm{M}$$(i=1,2)$ .

are open, there exists alarge integer $m’=m’(r)>0$ such that if$\rho(\nu,\mu_{\overline{x}_{0}})<$(Mf)), then$\nu(U(\tilde{y}.\cdot, r))>\delta=6(\mathrm{r})$By Lemma 4we can find $\epsilon’=\epsilon’(r)>0$ such that for$(i=1,2)$ .

(5)$\tilde{x}\in \mathrm{U}(\mathrm{x}0, \epsilon’)\cap F_{\epsilon}$$\rho(\mu_{\overline{x}},\mu_{\overline{x}_{0}})<1/2m’=1/2m’(r)$.

(6)

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

.

60
Let

$\xi$

be as in Lemma 2and put$\}$$B_{m}(n)= \{\tilde{x}\in M_{f}|\rho(\mu^{k/2]}\frac{[}{x}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}((\overline{f}^{-k+[k/2]}’\xi)(\overline{f}^{-k}\overline{x}))\mu_{\overline{x}})<1/m,<1/m$$(k\geq n)$

for $n,m\geq 1$ . Then $B_{m}(n)\subset Bm(n+1)$ and $\mu(\bigcup_{n=0}^{\infty}B_{m}(n))=1$ by (1) and (4), and sosuch that $\mu(B_{m}(n_{m}))\geq 1-1/2^{m+1}(m\geq 1)$ .
there exists an increasing sequence$m\geq 1$ , we can find $D_{m}\in B$ with $\mu(D_{m})\geq 1-2^{-m/2}$
for
Since
satisfying$\{n_{m}\}$$\mu(\bigcap_{k=m}^{\infty}B_{k}(n_{k}))\geq 1-1/2^{m}$$\mu_{\overline{x}}(\bigcap_{k=m}^{\infty}B_{k}(n_{k}))\geq 1-2^{-m/2}$$(\tilde{x}\in D_{m})$.

(7)

For $0<r<\tau/2$ we put$K_{\mathrm{r}}= \cap\cup k=1m=k\infty\infty(_{n=0}^{\infty}\cap\bigcup_{\ell=n}^{\infty}\tilde{f}^{-\ell}(U(\tilde{x}_{0},\epsilon’(r))\cap F_{\epsilon}\cap D_{m}))$.

for $m$ sufficiently large,
Since
Therefore,
obtain the conclusionto.
of$\mu(K_{r})=1(0<r<\tau/2)$
ergodicity
theby
we have.
forset-chaotic
is
that
to
show
it suffices
and
open
$(r
=1/n,n\geq
1)$
sets
nonempty
suppose
that
and
fix
To do thissatisfy$(j=1,2)$ .
,$\mu(U(\tilde{x}_{0},\epsilon’(r))\cap F_{\epsilon}\cap D_{m})\geq\mu(U(\tilde{x}_{0},\epsilon’(r))\cap F_{\epsilon})-2^{-m/2}>0$$\mu$$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\mu_{\overline{x}})$$\overline{x}\in K=\bigcap_{n\geq 1}K_{1/n}$$\mathrm{a}*$$U_{1}$$\tilde{x}\in K_{r}$$U_{1}\cap U_{2}\neq 0$$U_{2}$$U_{j}\cap \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\mu_{\overline{x}})\neq\emptyset$

Choose $m_{0}>0$ with
$0<2^{-m_{0}/2}< \min\{\mu_{\overline{x}}(U_{j}) :Since$\overline{x}\in K_{r}$integers$\{\ell_{k}\}_{k}$j=1,2\}$

and $m_{0}\geq 2m’$ .

, there exist $m_{1}>m_{0}$ and asequence of positive, by the definition ofsuch thatwith$K_{r}$$\ell_{k}>n_{k}$$\tilde{f}^{\ell_{k}}(\tilde{x})\in U(\overline{x}_{0},\epsilon’(r))\cap F_{\epsilon}\cap D_{m_{1}}$$(k\geq 1)$.

(8)

Thus, by (3) and (7) we have$\mu_{\overline{x}}(\overline{f}^{-\ell_{k}}(B_{k}(n_{k})))$$\geq\mu_{\overline{x}}(\tilde{f}^{-\ell_{k}}(\bigcap_{k=m_{1}}^{\infty}B_{k}(n_{k})))$$= \mu_{\overline{f}^{\ell_{k}}(\overline{x})}(\bigcap_{k=m_{1}}^{\infty}B_{k}(n_{k}))$$\geq 1-2^{-m_{1}/2}\geq 1-2^{-m\mathrm{o}/2}$

and so$\mu_{\overline{x}}(U_{j}\cap\tilde{f}^{-\ell_{k}}(B_{k}(n_{k})))\geq\mu_{\overline{x}}^{u}(U_{j})-2^{-m\mathrm{o}/2}>0$$(k\geq m_{1})$,. Therefore we can choose$\overline{z}_{j}=\overline{z}_{j}(k)\in U_{j}\cap\overline{f}^{-\ell_{k}}(B_{k}(n_{k}))\cap\eta(\overline{x})$for $j=1,2$ andSince$k\geq m_{1}$.

, we have$\overline{f}^{\ell_{k}}(\tilde{z}_{j})\in B_{k}(n_{k})\cap\overline{f}^{\ell_{k}}(\eta(\tilde{x}))\subset B_{k}(\ell_{k})\cap\eta(\tilde{f}^{\ell_{k}}(\overline{x}))$$\rho(\mu_{(_{j}^{\frac{]}{z}})}^{1_{\frac{\ell}{f}p_{k}}},\mu_{\overline{f}^{e_{k(\overline{x})}}})k/2=\rho(\mu^{1_{\frac{\ell}{f}\ell_{k(_{j}^{\frac{]}{z}})}}}, \mu_{\overline{f}^{\ell_{k}}(\overline{z}_{\mathrm{j}})})k/2<1/k\leq 1/m_{0}\leq 1/2m’$$\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}((\tilde{f}^{-\ell_{k}+[\ell_{k}/2]}\xi)(\tilde{z}_{j}))<1/k$

for $j=1,2$ and $k\geq m_{1}$ . By use of (6) and (8)$\rho(\mu^{1_{\frac{\ell}{f}p_{k(_{j}^{\frac{]}{z}})}}},\mu_{\overline{x}0})k/2$$\leq\rho(\mu^{[_{k}/2}\frac{\ell}{f}\ell_{k(_{j}^{\frac{]}{z}})}’\mu_{\overline{f}^{\ell_{k}}(\overline{x})})+\rho(\mu_{\overline{f}^{p_{k}}(\overline{x})},\mu_{\overline{x}_{0}})$$<1/\mathrm{m},$and so$+1/2m’=1/m’$ ,$\mu_{\overline{z}_{\mathrm{j}}}^{-\ell_{k}+[\ell_{k}/2]}(\overline{f}^{-\ell_{k}}U(\tilde{y}.\cdot,r))=\mu_{p_{k}}^{[_{\frac{\ell}{f}}}(k/2U(\overline{y}_{\dot{l}}, r))(_{j}^{\frac{]}{z}})>\delta$by (5). Thus we have$(\overline{f}^{-\ell_{k}+[\ell_{k}/2]}\xi)(\tilde{z}_{j})\cap\overline{f}^{-\ell_{k}}U(\overline{y}.\cdot,r)\neq 0$

,

(9)

61
for $1\leq i,j\leq 2$ and $k\geq m_{1}$ . Since$\tilde{z}_{j}\in U_{j}$, by (9) we may assume$\tilde{z}_{j}\in(\overline{f}^{-\ell_{k}+[\ell_{k}/2]}\xi)(\overline{z}_{j})\subset U_{j}$

for

$k$

large enough. Therefore$U_{j}\cap\tilde{f}^{-\ell_{k}}U(\overline{y}\dot{.}, r)\supset(\overline{f}^{-\ell_{k}+[\ell_{k}/2]}\xi)(\tilde{z}_{\mathrm{j}})\cap\tilde{f}^{-\ell_{k}}U(\tilde{y}_{\dot{1}}, r)\neq\emptyset$

for $1\leq i,j\leq 2$ and
Now we take$k$large enough.

$b_{:,j}=b_{i,\mathrm{j}}(k)\in U_{j}\cap\overline{f}^{-\ell_{k}}U(\tilde{y}_{\dot{1}}, r)$$b_{:,\mathrm{j}}\in U_{\mathrm{j}}$$(1\leq i,j\leq 2)$

for $1\leq i,j\leq 2$ and then,and$d(\tilde{f}^{\ell_{k}}(b_{1,1}),\overline{f}^{\ell_{k}}(b_{2,2}))>d(U(\tilde{y}_{1}, r),$ $U(\overline{y}_{2},r))>\tau$$d(\overline{f}^{\ell_{k}}(b_{1,1}),\overline{f}^{\ell_{k}}(b_{1,2}))\leq \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}(\mathrm{U}(\mathrm{y}\mathrm{u}\mathrm{r}))=2r=2/n$This implies that$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(/\mathrm{X}\mathrm{i})$is$\mathrm{a}*$-chaotic set for$\tilde{x}\in K=\bigcap_{n\geq 1}K_{1/n}$.

.
$\square$

References[1] N. Aoki and K. Hiraide, Topological theory of dynamical systems. Recent advances,
North-Holland Mathematical Library, 52. North-Holland Publishing Co., Amsterdam,

1994.

[2] L.S. Block and W.A. Coppel, Dynamics in One Dimension, Lecture Notes in Math.
1513, Springer, Berlin, 1992.
[3] C. Castaing and M. Valadier, Convex analysis and measurable multifunctions, Lecture
Notes in Mathematics, Vol. 580. Springer-Verlag, Berlin-New York, 1977.
[4] Y.M. Chung, Shadowing property of non-invertible maps with hyperbolic measures,Tokyo J. Math. 22 (1999), no. 1, 145-166.

[5] R. Devaney, An Introduction to Chaotic Dynamical Systems, Addison-Wesley, Reading,MA, 2nd ed., 1989.
[6] A. Fathi, M.R. Herman and -C. Yoccoz, A proof of Pesin’s stable manifold theorem,
Geometric Dynamics (J. Palis, Jr., Ed.), Lecture Notes in Math., 1007 Springer, Berlin,
1983, 177-215.
$\mathrm{J}$[7] M. Hata, Scrambled sets on compact metric spaces, J. Math. Kyoto Univ., 24-4 (1984),

689-698.

[8] H. Kato, Chaotic continua of (continuum-wise) expansive homeomorphisms and chaosin the sense of Li and Yorke, Fund. Math., 145 (1994), 261-279.
[9] H. Kato, Everywhere chaotic homeomorphisms on manifolds and -dimensional Menger
manifolds, Topology AppL, 72 (1996), 1-17.
$k$[10] H. Kato, On scrambled sets and a theorem of Kuratowski on independent sets, Proc.
Amer. Math. Soc, 126 (1998), 2151-2157.

[11] A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Publ.
Math. I.H.E.S., 51 (1980), 137-174.
[12] K. Kuratowski, Topology, Academic Press, New York, 1968.
[13] F. Ledrappier and J.M. Strelcyn, A proof of the estimation from below in Pesin’s entropy
fomula, Ergod. Th. &Dynam. Sys., 2(1982), 203-219

62[14] F. Ledrappier and L.S. Young, The metric entropy of diffeomorphisms, Ann. of Math.,
122 (1985), 509-574.

[15] T.Y. Li and J.A. Yorke, Period three implies chaos, Amer. Math. Monthly, 82 (1975),

985-992.
[16] P.D. Liu and M. Qian, Smooth ergodic theory of random dynamical systems. Lecture
Notes in Mathematics, 1606. Springer-Verlag, Berlin, 1995.
[17] P.D. Liu, Pesin’s entropy formula for endomorphisms, Nagoya Math. J. 150 (1998),

197-209.

[18] R.

, Ergodic Theory and Differentiate Dynamics, Springer, Berlin, 1987.

$\mathrm{M}\mathrm{a}\overline{\mathrm{n}}\text{\’{e}}$[19] F.R. Marotto, Snap-back repellers imply chaos in(1978), 199-223.

$\mathrm{R}^{\ltimes}$, J. Math. Analysis and Appl., 63[20] M. Misiurewicz and J. Sm\’ital, Smooth chaotic maps with zero topological entropy, Ergod. Th. &Dynam. Sys., 8(1988), 421-424.
[21] V.I. Oseledec, A multiplicative ergodic theorem, Lyapunov characteristic numbers for
dynamical systems, Trans. Mosc. Math. Soc, 19 (1968), 197-231.
[22] Y.B. Pesin, Families of invariant manifolds corresponding to non-zero characteristic
exponents, Math, of the USSR, Izvestija, 10 (1978), 1261-1305.

[23] Y.B. Pesin, Characteristic Lyapunov $e\eta onen\hslash$ and smooth ergodic theory, Russ. Math.
Surveys, 32:4 (1977), 55-114.
[24] C. Pugh and M. Shub, Ergodic attractors, Trans. Amer. Math. Soc., 312 (1989), 1-54.
[25] M. Qian and Z. Zhang, Ergodic theory for axiom A endomorphisms, Ergod. Th. andDynam. Sys. 15 (1995), 161-174.
[26] V.A. Rohlin, On the fundamental ideas of measure theory, A.M.S. TVansl., (1) 10 (1962),

1-52.

[27] V.A. Rohlin, Lectures on the theory of entropy of transformations with invariant measures, Russ. Math. Surveys, 22:5(1967), 1-54.
[28] D. Ruelle, An inequality for the entropy of differentiable maps, Bol. Soc. Bras. Math.,
9 (1978), 83-87.
[29] D. Ruelle and M. Shub, Stable manifolds for maps, Global theory of dynamical systems(Proc. Internat. Conf., Northwestern Univ., Evanston, 111., 1979), pp. 389392, Lecture
Notes in Math., 819, Springer, Berlin, 1980.

[30] D. Ruelle, Characteristic exponents and invariant manifolds in Hilbert space, Ann. of
Math. (2) 115 (1982), no. 2, 243-290.
[31] N. Sumi, Stable and unstable manifolds of diffeomorphisms with positive entropy, Singular phenomena of dynamical systems (Kyoto, 1999). S\={u}rihisekikenky\={u}sho Kokyuroku
No. 1118 (1999), 13-26.
[32] P. Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, 79.
Springer-Verlag, New York-Berlin, 1982.

[33] P. Walters, A dynamical proof of the miltiplicative ergodic theorem, Trans. Amer. Math.
Soc. 335 (1993), no. 1, 245257.
[34] J. Xiong and Z. Yang, Chaos caused by a topologically miing map, World Scientific,
Advanced Series in Dynamical Systems Vol9 (1991), 550-572