数理解析研究所講究録
1118 巻 1999 年 27-30

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MULTIFRACTAL ANALYSIS FOR ONE-DIMENSIONAL MAPS
WITH A NEUTRAL FIXED POINT(中立不動点をもつ力学系のマルチフラクタル解析)
KENTARO NAKAISHI(中石健太郎)
GRADUATE SCHOOL OF MATHEMATICAL SCIENCES,
UNIVERSITY OF TOKYO(東京大学)

1. INTRODUCTION & RESULTS

Recent developments of computers have made us possible to see fractal sets
casually. If you have any experience of drawing fractal pictures by computer, youmust have noticed that some points are easy to draw while others are not. Thismeans that an invariant set does not necessarily have uniform structure from thedynamical point of view. One of the motivations for the multifractal analysis is touncover this hidden structure.
Given a measure on a set $X$ , a typical multifractal object is the Hausdorff
dimension of points with the same scaling ;$\mu$$\alpha$.

$f_{\mu}( \alpha):=\mathrm{H}\mathrm{D}\{x\in X|\lim_{rarrow 0}\frac{\log\mu(B(x,r))}{\log r}=\alpha\}$For conformal hyperbolic systems with Gibbs measures , the dimension spectrum
is real analytic. On the other hand, based on the analogy of statistical mechanics, it is expected that spectra of non-hyperbolic systems
fail to be real analytic and that the point of phase transition has some characteristics. In this work, we present some results along this line.
Let : $[0,1]arrow[0,1]$ be a piecewise monotone interval map with the following
properties;(A1). each of $f|_{(0,c}$),) is strictly monotone and extends to afunction
andwith $f_{1}[0, c]=f_{2}[c, 1]=[0,1]$ .
(A2) $f_{1}(0)=0$ andexcept at $z=0$ .
is(A3) $Df_{1}(0)=1$ and $Df_{1}(z)>1$ for $z\in[0, c]$ ,(A4) $|Df_{2}(z)|>1$ on
and $|Df_{2}(Z)|\geq 1$ otherwise.
(A5)
has an asymptotic expansion around of the form$\mu$$\alpha\mapsto f_{\mu}(\alpha)$$(\mathrm{P}\mathrm{e}\sin- \mathrm{w}\mathrm{e}\mathrm{i}\mathrm{S}\mathrm{s}[2])$$f$$0<\exists c<1\mathrm{s}.\mathrm{t}$$C^{1}$$f|_{(^{C,1}}$$f_{2}$$f_{1}$$f_{1}$$C^{2}$$f_{2}^{-1}([c, 1])$

$D^{2}f_{1}$$0$$D^{2}f(z)=b_{Z^{\mathcal{T}}}-1+o(z^{\tau-}1),$$b>0,$$\tau>0$.

Example 1.1 (Farey map). $c=1/2$ ,$f_{1}(z)=Z/(1-Z),$ $f2(Z)=(1-Z)/z$

Let

$u$

be the measure of maximal entropy for .
$f$

Definition 1.1. For$\alpha\in \mathbb{R}^{+}$

define$f_{u}( \alpha)=\mathrm{H}\mathrm{D}\{z\in[0,1];\lim_{rarrow 0}\frac{\log u(B(_{Z},r))}{\log r}=\alpha\}$,$l( \alpha)=\mathrm{H}\mathrm{D}\{z\in[0,1];\lim_{arrow n\infty}\frac{1}{n}\log|Df^{n}(z)|=\alpha\}$where HD stands for Hausdorff dimension.
the Lyapunov spectrum for .
of and$f_{u}(\alpha)$$u$$l_{u}(\alpha)$$f$is called the dimension spectrum

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Main Theorem.

$\exists T(q)$.
real analytic on$\lim_{qarrow 0}DT(q)$ exists,
$q>0$
and
for
unbounded. $(f_{u}(\alpha), T(q))$ forms a Legendre transform$\mathbb{R}^{+}s.t$(a) $D^{2}T(q)>0,$ $T(\mathrm{O})=1$(b) The domain of$f_{u}(\alpha)i\mathit{8}$$pair_{i}$$f_{u}(\alpha(q))=\tau(q)+q\alpha(q)$$(c)$

where $\alpha(q)=-DT(q)>0$ ,is finite then $f_{u}(\alpha)=1$ for every
If$\alpha(0+)$$(d)l(\alpha)=f_{u}(\log 2/\alpha)$$\alpha\geq\alpha(0+)$.

,

Corollary. The followings are equivalent;(a) $\alpha(0)=\infty$ .
(b)
are real analytic.
and(c) The absolutely continuous invariant measure is infinite.
(d) The Lyapunov exponent with respect to the absolutely continuous invariant
measure is .
$f_{u}(\alpha)$$l_{u}(\alpha)$$\mathit{0}$FIGURE 1. Typical spectral shapes

2. How TO PROVE
Put $J=[c, 1],$and let $I_{k}=f_{1}^{-}k(J)$ for all $k\geq 0$ . Define the subintervals$J_{k}=f_{2}^{-1}(I_{k}-1)$. The next lemma is simple but essential through thisof bywork.
$I_{\infty}=\{0\}$$\{J_{k}\}$$J$

Lemma 2.1. For arbitrarily small$\epsilon>0$

there exist constants$C_{1}k^{(\epsilon}1-)(1+\tau)/\tau\leq Df^{k}(_{Z})\leq C_{2}k^{(1}+\epsilon)(1+\tau)/\tau$for all$k\in \mathrm{N}$and all$z\in I_{k}$.

$C_{1},$ $C_{2}$so that

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It can be shown that the induced map $F$ on
is an expanding Markov map
with a countable partition and, using the lemma above,that $F$ has bounded distortion. Thus we can apply to $F$ the techniques of the thermodynamic formalism.
Define parameterized weight functions
by$[\mathrm{c}, 1]$$\phi_{s,q}$$\phi_{s,q}(z)=q\log\psi-s\log|DF(z)|$where $\psi(z)=2^{-k},$$z\in \mathrm{i}\mathrm{n}\mathrm{t}J_{k}$.

Theorem 2.2 (Walters). For eachinvariant ergodic probability measure$con\mathit{8}tant\Delta(n)\geq 1$ satisfying

there exists an Fso that for each $n>0$ there is a$(s, q)\in \mathbb{R}\cross \mathbb{R}^{+}\cup D\cross\{\mathrm{O}\}$$\nu_{s,q}$

on

$J$

$\lim_{narrow\infty}\log\Delta(n)/\log\sup_{K\in \mathcal{P}^{n}}|K|arrow 0$

and$\nu_{s,q}(F^{n}A)\Delta(n)\sim\int_{A}\exp(nP(s,q)-\phi(s,qn))d\nu_{s},q$whenever$F^{n}$: $Aarrow F^{n}A$ is invertible.

Definition 2.1. We call the invariant measure above the equilibrium state for.
and the number $P(s, q)=\log\lambda(s, q)$ the pressure for$\phi_{s,q}$$\phi_{s,q}$

The next gives the definition of $T(q)$ ( a generalization of Bowen’s formula).
Lemma 2.3. For each$q\in \mathbb{R}^{+}\cup\{0\}$, the function $srightarrow P(s, q)$ is convex, strictlydecreasing(hence continuous) and has a unique zero $T(q)$ .

Consider the one-parameter family of mappings. Letfor. Since is ergodic for each $q>0$ , there exists
be the equilibrium state fora positive constantsuch that for almost every $z\in J$$q\in \mathbb{R}^{+}\cup\{0\}$$\phi_{\tau(q),q}$$\phi_{\tau(q),q}$$\nu_{q}$$\nu_{q}$$\alpha=\alpha(q)$$\sum_{l=0}^{k-1}-\log\psi(F_{Z}^{l})/\sum_{l=0}\log|DF(F_{Z}l)|k-1arrow\alpha(q)=-\int\log\psi d\nu q/\int\log|DF|d\nu q$.

Then define the set$K( \delta)=\{z\in J\backslash J_{i}nv;\lim_{karrow\infty}\frac{\sum_{\iota=}^{k-1}0-\log\psi(F\iota z)}{\sum_{l=0}^{k-1}\log|DF(F^{l}Z)|}=\delta\}$and denote the Hausdorff dimension of $X$ (set or measure) by HD (X).
Proposition 2.4. For$q\in \mathbb{R}^{+}$$\mathrm{H}\mathrm{D}(\nu_{q})=\mathrm{H}\mathrm{D}(K(\alpha(q)))=T(q)+q\alpha(q)$The next lemma relates the original map
formalism.
Lemma 2.5. For(a)$z\in J\backslash J_{inv}$$\lim_{rarrow 0}=\underline{\log u(B(z,r))}\alpha$$\log r$$f$.

with the induced $F$ in the multifractal

each of the following three implies the other two:.

(b)$\lim_{narrow\infty}\frac{-\sum_{l}^{n-1}=0\mathrm{g}\psi 1\mathrm{o}(F^{l}z)}{\sum_{l=0}^{n-1}\log|DF(F^{\iota}z)|}=\alpha$(c)$\lim_{narrow\infty}\frac{1}{n}\log|Dfn(z)|=\frac{\log 2}{\alpha}$.

.

So far we have shown how to prove the main thoerem except the analyticity of, which is closely connected to Corollary.

$T(q)$

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Proposition 2.6.

(a) $T(q)$ is real analytic for $q>0$ and satisfies.

$DT(q)= \frac{\int\log\psi(Z)d\nu_{q}(Z)}{\int\log|DF(Z)|d\nu(qZ)}=-\alpha(q)$(b) If $\Psi:=\log\psi_{-}D\tau(q)\log|DF|$ then$D^{2}T(q)=( \int\log|DF(z)|d\nu_{q}(z)\mathrm{I}^{-1}\sigma(2\Psi)$

where$\sigma^{2}(\Psi)=\frac{d^{2}}{ds^{2}}P(T(q)+SDT(q), q+S))|_{\mathit{8}=0}$.

The formula in (a) leads to the proof of Corollary.

3. How TO FILL IN THE DETAILSWe refer to [5] for any reader who is interested in the details. See [1] also forthe general background of multifractal analysis. The paper [4] contains number
theoretical applications and shares some results with this work, but its proof seems
to be missing some point.
REFERENCES
Views and Applica’[1] Y. Pesin,Dimension Theory in Dynamicaltions,The University of Chicago Press,Chicago,1998.
[2] Y. Pesin and H. Weiss, A multifractal analysis of equilibrium measures for conformal expanding maps and Moran-like geometric constructions, J. Stat. Phys. 86 (1997), 233-275.
[3] –, The multifractal analysis of Gibbs measures: motivation,mathematical foundatioin
and examples, Chaos 7 (1997), 89-106.
[4] M. Pollicott and H. Weiss,Multifractal analysis for the continued fraction and MannevillePomeau transformations and applications to Diophantine approximation,preprint.
[5] K. Nakaishi, Multifractal formalism for some parabolic maps, Ergod. Th. b Dynam. Sys. to$Sy_{S\iota em}S:c_{\mathit{0}n\mathrm{f}y}empora$appear.