数理解析研究所講究録
1042 巻 1998 年 143-146

143

Structures of Lyapunov Regular Sets withNon-Zero Exponents
Masahiro KURATA
Nagoya Institute of Technology

1

Inroduction

Let $M$ be a closed manifold with a Riemannian metric and $f;Marrow M$ bea -diffeomorphism. For $\mu>0$ and $1\leq k\leq\dim M-1$ , we denote by
the setof Lyapunov regular points such that$C^{2}$$\lambda,$$\Lambda=\Lambda(-\mu, \lambda, k)$$x$$\mathrm{c}\mathrm{o}\mathrm{n}\dot{\mathrm{s}}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{g}$1. Lyapunov exponents of$x$

are less$\mathrm{t}\mathrm{h}\mathrm{a}\mathrm{n}-\mu$

or greater than ,$\lambda$

2. the dimension of the fiber of stable bundle $E^{s}(x)$ equals .
$k$Without loss of generality, we may assume that A has a dense orbit.
Inthis note we construct a “symbolic dynamics ”that represents , and study thestructure of A in the case when A is a fractal set.
$\Lambda$

2

symbolic dynamics of$\Lambda$

The set A is represented by a “symbolic dynamics ”as follows.
I Let
be a family of setsof words of length $n+1$ suchthat$\mathcal{W}=\{W_{n}\}_{n\geq 0}$$W_{n}$

1. There are symbols$\{B_{1}, \cdots, B_{p}.’C_{1}, \cdots., C_{q}\}$, and for any $n\geq 0$$W_{n}\subset\{B_{1}, \cdots, B_{p}, C_{1}, \cdots, C_{q}\}^{n+1}$2.

$(\alpha_{0}, \cdots , \alpha_{n})\in W_{n}$3. If

implies$\alpha_{0},$$\alpha_{n}\in\{B_{1}, \cdots , B_{p}\}$

4. for any$B_{i},$$\alpha_{0=}Bi,$$B_{j}(1\leq\dot{i},j\leq p)$$\alpha_{n.j}=B$,, there are $n\geq$.

.

For a family of sets of wordsof shift (not necessarily subshift)
II

2.

$1$

,

$\alpha_{n}=\beta_{0}..$

and’

then

$(\alpha_{0}, \cdots, \alpha_{n})\in W_{n}$with

as above, we define a subset
as follows:$\mathcal{W}=\{W_{n}\}_{n\geq 0}$$\Sigma=\Sigma(\mathcal{W})$1.

an.d$(\alpha_{0}, \cdots , \alpha_{n})\in W_{n},$ $(\beta_{0}, \cdots, \beta_{m})\in W_{m}$$(\alpha_{0}, \cdots, \alpha_{n}, \beta_{1}, \cdots, \beta_{m})\in W_{n+m}$,$\Sigma=\Sigma(\mathcal{W})\subset\{B_{1}, \cdots, B_{p}, C_{1,}.\cdots, C_{q}\}^{\mathbb{Z}}$$\Sigma$is generated by(a)$\mathcal{W}=\{\mathcal{W}_{n}\}_{n\geq 0}$$\underline{\alpha}=(\alpha_{n})_{n\in \mathbb{Z}}\in\Sigma$

such that, that is,if and only if for any $N>0$ there.are $m,$ $n\geq N$$(\alpha_{-m}, \alpha_{-m+1}, \cdots, \alpha_{0}, \cdots , \alpha_{n})\in W_{m+n}$

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(b)

$\alpha_{0}\in\{B_{\mathrm{O}}, \cdots, B_{p}\}$

III We denote by, where, is dea collection offined in. I an. And $an$ equivalence
defined in the disjoint union
as follows:
iffor
anthe, where is the shift map.
IV For the quotient space, a shift ma.p :is defined
as follows:for, we have,withwhereis given by.
$\Sigma$$\mathcal{W}=\{W_{n}\}_{n\geq 0}$$\Sigma=\Sigma(\mathcal{W})$$\mathrm{d}\mathrm{I}\mathrm{I}$$\mathrm{r}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}_{\mathrm{o}\mathrm{n}}\sim \mathrm{i}\mathrm{s}$$\cup\Sigma(\mathcal{W})$$\underline{a}\in\Sigma,\underline{b}\in\Sigma’$$\underline{a}\sim\underline{b}$$n$$\mathrm{d}\sigma^{n}(\underline{a})\in\Sigma,$$\sigma^{n}(\underline{a})\sim\sigma^{n}(\underline{b})$$\sigma^{r\iota}(\underline{b})\in\Sigma’$$\sigma$$\tilde{\Sigma}arrow\tilde{\Sigma}$$\overline{\Sigma}=\cup\Sigma(\mathcal{W})/\sim$$\underline{\alpha}=(\alpha_{n})_{n}\in\Sigma$$\alpha 0,$$\tilde{\sigma}$$\alpha_{k}\in\{B_{1}, \cdots, B_{p}\}$$\tilde{\alpha}^{k}[\underline{\alpha}]=[\underline{\beta}]$$\beta_{n}=\alpha_{n+k}$$\underline{\beta}=(\beta_{n})_{n}\in\Sigma$

With the notations as above, we have the following.

Theorem 2.1. For a Lyapunov regular set
1. a collection$\Sigma=\{\Sigma(\mathcal{W})\}$

2. an equivalence$\tilde{\sigma}$:$\Lambda=\Lambda(-\mu, \lambda, k)$of countable subsets of shifts,

3. a collection of maps:
$relat\dot{i}on\sim$with the equivalence,$\Psi=\Psi_{\Sigma}$

A (for$\Sigmaarrow$$\Sigma(\mathcal{W})$,and the shifl map$relat\dot{i}on\sim on\cup\Sigma(\mathcal{W})$$\tilde{\Sigma}=\cup\Sigma(\mathcal{W})/\simarrow\tilde{\Sigma}$, there are$\Sigma\in\Sigma$) which is compatible

such that the map$\tilde{\Psi}$

induced from$\{\Psi=\Psi_{\Sigma}|\Sigma\in\Sigma\}$:$\tilde{\Sigma}arrow$A

is surjective and the diagram$\tilde{\Sigma}rightarrow\overline{\sigma}\tilde{\Sigma}$$\overline{\Psi}\downarrow$$\downarrow\overline{\Phi}$$\Lambdaarrow f|\Lambda$

A

is commutative.

Remark 2.1. Letchoose the symbols
that$\epsilon$

be an arbitrary positive number. In Theorem2.1. we may
suchof any and maps$\{B_{1}, \cdots , B_{p}, C_{1}, \cdot.

t, C_{q}\}$$\Psi=\Psi_{\Sigma}$$\mathcal{W}$

1. $p=1$ ,2. diam

3. for$\Psi(\Sigma)<\epsilon$for$\Sigma=\Sigma(\mathcal{W})\in\Sigma$,$x=\Psi((\alpha_{n})_{n})\in\Psi(\mathcal{W})$$||\mathrm{T}f^{n}|\mathrm{E}^{S}(X)||<\exp(-\mu n)$$||\mathrm{T}f^{-n}|\mathrm{E}^{u}(x)||<\exp(-\lambda n)$$\dot{i}f\alpha_{0}=\alpha_{n}=B_{1}$if,$\alpha_{-n}=\alpha_{0}=B_{1}$.

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3

Locally self-similarity with countable contrac-tions
be a Lyapunov regular set. In the sequel we assume thatare given as in:, and maps
section 2 and satisfy Remark2.1.
Then A is a countable union of closed sets:
Let.

$\Lambda=$$\Lambda(-\mu, \lambda, k)$$\Sigmaarrow\Lambda(\Sigma\in\Sigma)$$\Psi=\Psi_{\Sigma}$$\Sigma=\{\Sigma\},\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\Sigma=\Sigma(\mathcal{W})$$\Lambda=\bigcup_{\Sigma\Sigma\in}\Psi(\Sigma)$.

.
In this section we consider the structure ofis,,thatbe the set of generators ofLet$\Psi(\Sigma)$$\mathcal{W}$$G_{k}(\mathcal{W})$,$G_{k}(\mathcal{W})=\{(a\mathrm{o}, \cdots.a_{k})\in W_{k}|a_{0}=a_{k}=B_{1}, a_{i}\in\{C_{1}, \cdots, C_{q}\}1\leq\dot{i}\leq k-1\}$$G(\mathcal{W})=\cup c_{k}k(\mathcal{W})$For$\mathrm{a}=$.

, the right contraction$(a_{0}, \cdots , a_{k})\in G_{(}\mathcal{W})$$R(\mathrm{a}):\Psi(\Sigma)arrow\Psi(\Sigma)$is defined by$R(\mathrm{a})(\Psi((\alpha_{n})n)=\Psi((\beta_{n})_{n})$for$(\alpha_{n})_{n}\in\Sigma$

,

where
$\alpha_{n-k}$$\beta_{n}=\{$ $a_{n}$$\alpha_{n}$,,,$k\leq n$,$0\leq n\leq k$$n\leq 0$,.

Similarly the left contraction$L(\mathrm{a}):\Psi(\Sigma)arrow\Psi(\Sigma)$is defined by$L(\mathrm{a})(\Psi((\alpha_{n})n)=\Psi((\beta_{n})_{n})$for$(\alpha_{n})_{n}\in\Sigma$

,

where
$\alpha_{n}$,$\beta_{n}=\{$ $a_{n+k}$$\alpha_{n+k}$$0\leq n$,,,$-k\leq n\leq 0$ ,$n\leq-k$ .

Then we have the following.

Proposition 3.1. The set;
maps$\Psi(\Sigma)$

is a countable union of images of$\Psi(\Sigma)$$R(\mathrm{a})L(\mathrm{b})$$\Psi(\Sigma)=\mathrm{a},\mathrm{b}\in G(\bigcup_{w)}R(\mathrm{a})L(\mathrm{b})(\Psi(\Sigma))$

If the map$\Psi=\Psi_{\Sigma}$:$\Sigmaarrow$

A is injective, then for any$\mathrm{a},$.
$\mathrm{b}\in G(\mathcal{W})$the map$L(\mathrm{b})R(\mathrm{a})=R(\mathrm{a})L(\mathrm{b}):\Psi(\Sigma)arrow\Sigma(\Sigma)$

is a contraction. And$\Psi(\Sigma)$is self-similar by countable contractions.

by1464$\mathrm{H}\mathrm{a}\mathrm{u}\mathrm{s}\mathrm{d}_{0}\mathrm{r}\mathrm{f}\mathrm{f}-$dimension of local stable manifolds

Form the propositions in the revious section,we have$\Psi(\Sigma)=\cup R(\mathrm{a}1)L(\mathrm{b}_{1})\mathrm{a}1,\mathrm{b}_{1}\in G(\mathcal{W})(\Psi(\Sigma))$$= \bigcup_{(\mathrm{a}_{2},\mathrm{b}_{2}\in Gw)}\mathrm{a}_{1},\mathrm{b}_{1}\in G(\mathcal{W}\cup R(\mathrm{a}2)L(\mathrm{b}_{2})R(\mathrm{a}1)L(\mathrm{b}_{1}))(\Psi(\Sigma))$$= \bigcup_{\mathrm{a}_{1},\mathrm{a}_{2\in}G(w)}\mathrm{b}_{1},\mathrm{b}_{2}\in G\cup R(\mathrm{a}2)R(\mathrm{a}1)L(\mathrm{b}_{2})L(\mathrm{b}_{1})(w)\backslash (\Psi(\Sigma))$$\cup$$=$$R(\underline{\mathrm{a}})L(\underline{\mathrm{b}})(\Psi(\Sigma))$,$\underline{\mathrm{a}},\underline{\mathrm{b}}\in G(W)^{\mathrm{N}}$where$\underline{\mathrm{a}}=\langle \mathrm{a}_{1},$an$\mathrm{a}_{2},$$\cdots),$,$\underline{\mathrm{b}}=(\mathrm{b}_{1}, \mathrm{b}_{2}, \cdots)\in G(\mathcal{W})^{\mathrm{N}}$$\mathrm{d}$$L(\underline{\mathrm{b}})(\Psi(\Sigma))=\cap L(\mathrm{b}_{N})\cdots L(\mathrm{b}1)(\Psi N\geq 1(\Sigma))$Besides the set
manifold and. Let Lip$L(\underline{\mathrm{b}})(\Psi(\Sigma))$$\Psi(\Sigma)$.

coincides with an intersection of a local unstable.

$(R(\mathrm{a})|L(\underline{\mathrm{b}})(\Psi(\Sigma)))$be the Lipshitz constant of the map$R(\mathrm{a}):L(\underline{\mathrm{b}})(\Psi(\Sigma))arrow L(\underline{\mathrm{b}})(\Psi(\Sigma))$

By choosing$\epsilon>0$.

.

in Remark2.1 sufficiently small, we have

Lip$(R(\mathrm{a})|L(\underline{\mathrm{b}})(\Psi\langle\Sigma)))<\exp(-\lambda n)$

Because the number of the elements of
implies the following.
Proposition 4.1. For$\underline{\mathrm{b}}\in G(\mathcal{W})^{\mathrm{N}}$$\sum_{\mathrm{a}\in c(w)}$Lip$G_{k}(\mathcal{W})$, there is.

is less than or equals$c(\underline{\mathrm{b}})>0$$q^{k-1}$, thissuch that$(R(\mathrm{a})|L(\underline{\mathrm{b}})(\Psi(\Sigma)))^{c}(\underline{\mathrm{b}})=1$.

The number
dominates the Hausdorff dimension of the intersection of
the local unstable manifold and:$c(\underline{\mathrm{b}})$$\Psi(\Sigma)$

Proposition 4.2. The Hausdorff dimension$c(\underline{\mathrm{b}})$$ofL(\underline{\mathrm{b}})(\Psi(\Sigma))$.

is less than or equals

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numbers for dynamical systems,
Math.soc.19 (1968),197231.
$r_{\mathrm{b}\mathrm{a}\mathrm{n}\mathrm{s}.\mathrm{M}_{\mathrm{o}\mathrm{S}}\mathrm{c}\mathrm{o}\mathrm{W}}$[2] Y.B.Pesin, Lyapunov characteristic exponent and smooth ergodic theory,
Russian Math.Surveys 32.
$(1977),\mathrm{n}\mathrm{o}.4,55- 114$