Lyapunov exponents, dual Lyapunov exponents, and multifractal ...

CHAOS

VOLUME 9, NUMBER 4

DECEMBER 1999

Lyapunov exponents, dual Lyapunov exponents, and multifractal analysis Aihua Fana) Faculte´ de Mathe´matiques et d’Informatique, Universite´ de Picardie, 33, Rue Saint Leu, 80039 Amiens and Analyse Harmonique, Universite´ de Paris Sud, 91405, Orsay, France

Yunping Jiangb) Department of Mathematics, Queens College of CUNY, Flushing, New York 11367 and Department of Mathematics, Graduate School of CUNY, New York, New York 10036

共Received 19 March 1999; accepted for publication 13 June 1999兲 It is shown that the multifractal property is shared by both Lyapunov exponents and dual Lyapunov exponents related to scaling functions of one-dimensional expanding folding maps. This reveals in a quantitative way the complexity of the dynamics determined by such maps. © 1999 American Institute of Physics. 关S1054-1500共99兲00204-9兴 pp. 8–9 for a more comprehensive definition兲. It has a root in physics, in particular, in the study of period doubling bifurcations in chaos 共see Refs. 3,4,5兲. Scaling functions are also defined for Markov maps of intervals in Ref. 1, p. 76. Although the results in this article can be proved for any Markov map of an interval, we will only state them for folding maps of an interval 共in order to have notational simplicity兲. Let I be the interval 关 ⫺1,1 兴 . Suppose f :I→R is a map. Let f n denote the n-time composition of f . Suppose,

One of the topics in dynamical systems is to study the geometric structure of some invariant set associated with a given dynamical system. The scaling function has been used to describe the geometric structure of the maximal invariant set of a certain dynamical system studied in this article. In many cases the maximal invariant set is a fractal. More interestingly, the graph of the scaling function in many cases also looks like a fractal. Therefore to analyze a certain fractal property of the maximal invariant set and the dual symbolic space where the scaling function lives becomes important. The multifractal analysis provides us with a tool to conduct this study. In this article, we use the Lyapunov exponent and the dual Lyapunov exponent for such a given dynamical system to give some multifractal decompositions for the maximal invariant set and the dual symbolic space. We then use Ruelle–Perron–Frobenius operators in the thermodynamical formalism to calculate the Hausdorff dimensions of the subsets in the multifractal decompositions. Our analysis says that the Lyapunov exponent and the dual Lyapunov exponent for a dynamical system studied in this article are indeed multifractal functions.

共i兲共ii兲共iii兲

f (⫺1)⫽ f (1)⫽⫺1 and f (0)⬎1, f 兩 [⫺1,0] is increasing and f 兩 [0,1] is decreasing, f is continuously differentiable on f ⫺1 (I) with nonzero derivatives 共at an endpoint of an interval we only consider one-side derivatives兲.

We call such a map a C 1 folding map. The maximal invariant set of f is ⬁

⌳⫽ 艚 f ⫺n 共 I 兲 . n⫽0

Our main concern is the dynamical system (⌳, f ). We say a folding map f is expanding if, furthermore 共iv兲 There are constants C⬎0 and ␭⬎1 such that

I. INTRODUCTION AND STATEMENTS OF MAIN RESULTS

兩共 f n 兲 ⬘ 共 x 兲兩 ⭓C␭ n ,

The graph of a scaling function for a folding map 共see Ref. 1, p. 115兲 looks like a fractal. However, the direct study of the fractal property of the graph is rather difficult. In this article, we study the dual Lyapunov exponent which we define as the ergodic mean value of the scaling function. We prove that the dual Lyapunov exponent is a multifractal function 共Theorem 2兲. We also study the Lyapunov exponent of the folding map and show that it is also a multifractal function 共Theorem 1兲. Scaling functions for expanding C 1⫹ ␣ folding maps are introduced in mathematics by Sullivan in Ref. 2 共see Ref. 1,

j⫽0

冊

By a theorem due to Mather 共see Ref. 6 for a proof兲, for an expanding map f , there exists a C 1 change of coordinate such that under the new coordinate we have 兩 f ⬘ (x) 兩 ⭓␭⬎1 (᭙x苸 f ⫺1 (I)). In the following, ‘‘expanding’’ will mean this last condition. We will denote by ␻ f ⬘ the modulus of continuity of f ⬘ on f ⫺1 (I) which is defined in the usual way,

␻ f ⬘ 共 r 兲 ⫽ sup 兩 f ⬘ 共 x 兲 ⫺ f ⬘ 共 y 兲兩 , 兩 x⫺y 兩 ⭐r

where x and y stay in a same component of f ⫺1 (I). We say that f ⬘ satisfies the Dini condition if

冕␻ 1

a兲

Electronic mail: [email protected] Electronic mail: [email protected]; Current address: Institute for Mathematical Research 共FIM兲, ETHZ, 8092 Zurich, Switzerland. Electronic mail: [email protected]

b兲

1054-1500/99/9(4)/849/5/$15.00

冉

n⫺1

᭙x苸艛 f ⫺ j 共 I 兲 .

0

f ⬘共 r 兲

dr ⬍⬁. r

We say that f ⬘ satisfies the strong Dini condition if 849

© 1999 American Institute of Physics

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satisfies the Dini condition. Note that s is a C 1 -invariant 共refer to Ref. 1兲. For a scaling function s defined on ⌺ * , we define

兩 log r 兩 ␻ f ⬘共 r 兲 dr⬍⬁. r

The Lyapunov exponent of (⌳, f ) at x苸⌳ is defined as

冏

1 dfn L 共 x 兲 ⫽ lim log 共x兲 dx n→⬁ n

冏

as the n-variation of s. We say a scaling function is of summable variation if

共if the limit exists兲. It follows from the ergodic theorem that L(x) is almost everywhere constant with respect to an f -invariant ergodic measure and the constant is the integral of log兩 f ⬘(x)兩 with respect to this measure 共called the Lyapunov exponent of the measure兲. Since there are a lot of invariant ergodic measures, we should get different constants. In this paper, we first describe more quantitatively how the Lyapunov exponents are different by discussing the sizes via Hausdorff dimensions of the following sets: E ␣ ⫽ 兵 x苸⌳:L 共 x 兲 ⫽ ␣ 其 . Our study of the Hausdorff dimension dimE␣ is based on transfer operators. For ␤ 苸R, consider the transfer operator, T ␤␺共 x 兲⫽

兩 f ⬘共 y 兲兩 ␤␺ 共 y 兲 , 兺 y苸 f (x) ⫺1

共 x苸I 兲 .

The operator T ␤ acts on the space C(I) of continuous functions. The logarithm of its spectral radius is denoted as P log兩 f ⬘兩(␤) or simply P( ␤ ) which is called the pressure function.7,8 Theorem 1: Suppose f is an expanding folding map whose derivative satisfies the strong Dini condition. Then P( ␤ ) is everywhere differentiable and if ␣ ⫽ P ⬘ ( ␤ ) for some ␤ , we have dim E ␣ ⫽

P 共 ␤ 兲 ⫺ ␣␤ . ␣

Now let us look at the scaling function s of (⌳, f ). It is defined on the dual Cantor set ⌺ * 共see Ref. 1, p. 9兲 which is composed of sequences of the form ␻ * ⫽•••i n •••i 1 i 0 and is equipped with the usual ultrametric. Let ␪ * be the shift on ⌺ * defined as ␪ * :•••i n •••i 1 i 0 哫•••i n •••i 1 . Instead of directly studying f , we shall study its inverses. Let g 0 and g 1 be the left and right inverse branches restricted on I of f . By the above hypothesis made on f , the maps g 0 :I→g 0 (I) and g 1 :I→g 1 (I) are C 1 diffeomorphisms. For a finite sequence ␻ n ⫽ j 0 i 1 ••• j n⫺1 共read from left to right兲 of symbols 兵 0,1 其 , let g ␻ n ⫽g j 0 ⴰg j 1 ⴰ•••ⴰg j n⫺1 ,

I ␻ n ⫽g ␻ n 共 I 兲 .

Let ␻ * n ⫽i n⫺1 •••i 0 denote the same ␻ n but read from right to left. The scaling function s of (⌳, f ) is defined as s 共 ␻ * 兲 ⫽ lim s 共 ␻ * n 兲, n→⬁

共 ␻ * ⫽•••i n⫺1 •••i 1 i 0 苸⌺ * 兲

共if the limit exists for every ␻ * ), where s 共 ␻ n* 兲 ⫽

⬘ varn 共 s 兲 ⫽sup兵兩 s 共 ␻ * 兲 ⫺s 共 ␻ * ⬘ 兲兩 : ␻ n* ⫽ ␻ * n 其

兩 I i n⫺1 •••i 1 i 0 兩兩 I i n⫺1 •••i 1 兩

( 兩 J 兩 denoting the length of an interval J). It will be seen 共Theorem 3兲 that the scaling function is well defined when f ⬘

⬁

兺

n⫽1

varn 共 s 兲 ⬍⬁.

Furthermore, we prove that s is of summable variation if f ⬘ satisfies the strong Dini condition 共Corollary 1兲. We define the dual Lyapunov exponent of (⌳, f ) at ␻ * 苸⌺ * as 1 L * 共 ␻ * 兲 ⫽⫺ lim n n→⬁

n⫺1

兺

k⫽0

log s 共 ␪ * k ␻ * 兲

共if the limit exists兲. Our purpose is to study the sizes of the following sets: E ␣* ⫽ 兵 ␻ * 苸⌺ * :L * 共 ␻ * 兲 ⫽ ␣ 其 . We prove the following theorem: Theorem 2: Suppose f is an expanding folding map whose derivative satisfies the strong Dini condition. If ␣ ⫽ P ⬘ ( ␤ ) for some ␤ , we have dim E ␣* ⫽

P 共 ␤ 兲 ⫺ ␣␤ . log 2

We call 艛 ␣ E ␣* 共resp. 艛 ␣ E ␣ ) the multifractal decomposition of ⌺ * 共resp. ⌳) with respect to dual Lyapunov exponents 共resp. Lyapunov exponents兲. They are C 1 invariants. Theorems 1 and 2 say that Hausdorff dimensions of E ␣* and E ␣ are well calculated, that is, the dual Lyapunov exponent and the Lyapunov exponent for an expanding C 1 folding map whose derivative satisfies the strong Dini condition are multifractal functions.

II. DINI CONDITION, STRONG DINI CONDITION, AND DISTORTION

For n⭓0, denote In ⫽ 兵 I ␻ n : ␻ n ⫽ j 0 ••• j n⫺1 苸兵 0,1 其 n 其 ,

␦ n ⫽max兩 I 兩 . I苸In

Consider the space ⌺ of infinite sequences ␻ ⫽ j 0 j 1 ••• j n⫺1 ••• of 0’s and 1’s. For ␻ ⫽ j 0 j 1 •••苸⌺ and an integer n ⭓1, we write ␻ n ⫽ j 0 ••• j n⫺1 . Let ␪ :⌺→⌺ be the shift map defined by ( ␪ ␻ ) n ⫽ j n⫹1 . Let u j :⌺→⌺ ( j⫽0,1) be the inverses of ␪ , which can be defined by u j ␻ ⫽ j ␻ . Lemma 1: Suppose f is a folding map and suppose limn→⬁ ␦n⫽0. Then, the limit

␲ 共 ␻ 兲 ⫽ lim g ␻ n 共 x 兲 ,

共 ᭙x苸I 兲

n→⬁

exists and is independent of x. The function ␲ :⌺→⌳ is continuous and

␲ⴰ␪⫽ f ⴰ␲,

␲ ⴰu j ⫽g j ⴰ ␲ 共 j⫽0,1 兲 .


Chaos, Vol. 9, No. 4, 1999

Exponents and multifractal analysis

Proof: Note that 兵 I ␻ n 其 is a sequence of embedded inter⬁ I ␻ ⫽ 兵¯x 其 for some vals whose length tends to zero. So, 艚 n⫽0 n

¯x . Since g ␻ (x)苸I ␻ 共for all x苸I), we get the existence of n n ¯ . Recall that d( ␻ , ␻ ⬘ )⫽2 ⫺n , where the limit and ␲ ( ␻ )⫽x ⬘ 其 . If d( ␻ , ␻ ⬘ )⫽2 ⫺n , we have n⫽sup兵 m: ␻ m ⫽ ␻ m

兺 ␻ f ⬘共 ␭ ⫺(n⫺i)兩 x⫺y 兩兲 ⭐ 冕0 ␻ f ⬘共 ␭ ⫺t兩 x⫺y 兩兲 dt i⫽0

n⫺1

⬁

⫽

for some z,z ⬘ 苸I. So, 兩 ␲ ( ␻ )⫺ ␲ ( ␻ ⬘ ) 兩 ⭐ 兩 I ␻ n 兩 ⭐ ␦ n . To get the last equalities in the lemma, it suffices to check them by using the definition of ␲ and the fact that f ⴰg j is the identity of the interval I for j⫽0 or 1. 䊐 It follows from the last lemma that we have the following relations: ⌳⫽ ␲ 共 ⌺ 兲 ,

Note that 兩 y n⫺i ⫺x n⫺i 兩 ⭐␭ ⫺(n⫺i) 兩 x⫺y 兩 . The last sum is bounded by

␲ 共 ␻ ⬘ 兲 ⫽g ␻ n 共 z ⬘ 兲

␲ 共 ␻ 兲 ⫽g ␻ n 共 z 兲 ,

⌳⫽g 0 共 ⌳ 兲艛g 1 共 ⌳ 兲 .

The first relation says that ⌳ is the image of ⌺ under ␲ . The second relation means that ⌳ is a self-similar set 共see Ref. 9兲. Lemma 2: Suppose f is a C 1 folding map. For any finite sequence ␻ n of 0’s and 1’s of length n, g ␻ n ⴰ f n is the identity on I ␻ , f n ⴰg ␻ n is the identity on I, and we have dfn 1 共 g 共 y 兲兲 ⫽ dx ␻ n g ␻⬘ 共 y 兲

t

f ⬘共 r 兲

0

r

冏冏 g ␻⬘ 共 x 兲 n

g ␻⬘ 共 y 兲 n

⭐exp关 C⍀ f ⬘ 共兩 x⫺y 兩兲兴 ,

冏冏冏 g ␻⬘ 共 y 兲 n

⫽

冏

共 f 兲 ⬘共 y n 兲 n

共 f n 兲 ⬘共 x n 兲

.

It follows that

冏冏冏兺

log

g ␻⬘ 共 x 兲 n

g ␻⬘ 共 y 兲 n

冏

n⫺1

⭐

i⫽0

1 ⭐ ␭ 1 ⭐ ␭

log兩 f ⬘ 兩共 y n⫺i 兲 ⫺log兩 f ⬘ 兩共 x n⫺i 兲

n⫺1

兺

i⫽0

兩 f ⬘ 共 y n⫺i 兲 ⫺ f ⬘ 共 x n⫺i 兲兩

n⫺1

兺

i⫽0

␻ f ⬘ 共 y n⫺i ⫺x n⫺i 兲 .

t

0

dt.

where C⬎0 is a constant. Proof: For ␻ * ⫽•••i n i n⫺1 •••i 0 苸⌺ * , ⫽i n⫺1 •••i 0 . If m⬎n, we have

* 兲 ⫺s 共 ␻ n* 兲兩 ⫽ 兩s共 ␻m

冏冏

冏

⫽

兩 I i m⫺1 •••i 1 i 0 兩兩 I i m⫺1 •••i 1 兩

⫺

兩 I i n⫺1 •••i 1 兩

兩 g i m⫺1 •••i n 共 I i n ⫺1•••i 1 兲兩兩 g i⬘

共 ␰ m,n 兲兩

兩 g i⬘

共 ␩ m,n 兲兩

m⫺1 •••i n

冏

兩 I i n⫺1 •••i 1 i 0 兩

兩 g i m⫺1 •••i n 共 I i n⫺1 •••i 1 i 0 兲兩

m⫺1 •••i n

␻* n

let

⫺

冏

⫺1 •

冏

兩 I i n⫺1 •••i 1 i 0 兩兩 I i n⫺1 •••i 1 兩

兩 I i n⫺1 •••i 1 i 0 兩兩 I i n⫺1 •••i 1 兩

,

where ␰ m,n and ␩ m,n are some two points in I i m⫺1 •••i 1 . By the above naive distortion lemma, we have

冏

兩 g i⬘

m⫺1 •••i n

共 ␰ m,n 兲

兩 g i⬘

共 ␩ m,n 兲 m⫺1 •••i n

冏

⫺1 ⭐C⍀ f ⬘ 共 ␰ ⫺ ␩ 兲 ⭐C⍀ f ⬘ 共 ␦ n 兲 ,

where ␰ ⫽ f m⫺n ( ␰ m,n ) and ␩ ⫽ f m⫺n ( ␩ m,n ) are some two points in I i n⫺1 •••i 1 . Thus we have proved that

where C⫽ (1/␭ log ␭)⬎0 (which is a constant independent of x, y, and ␻ n ). Proof: Let x n ⫽g ␻ n (x) and y n ⫽g ␻ n (y) and let x n⫺i ⫽ f i (x n ) and y n⫺i ⫽ f i (y n ) for 0⭐i⭐n. Note that x 0 ⫽x ⫽ f n (x n ),y 0 ⫽y⫽ f n (y n ). By Lemma 2, we have g ␻⬘ 共 x 兲 n

f ⬘共 t 兲

varn 共 s 兲 ⭐C⍀ f ⬘ 共 ␦ n 兲 ,

⫽

Lemma 3 共naive distortion兲: Suppose f is an expanding C 1 folding map whose derivative satisfies the Dini condition and has expanding constant ␭⬎1. Then for any x,y苸I and any ␻ n ⫽ j 0 j 1 ••• j n⫺1 , we have

兩 x⫺y 兩 ␻

Theorem 3: Suppose f is an expanding C folding map whose derivative satisfies the Dini condition. Then the scaling function s exists and satisfies

共 ᭙y苸I 兲 .

dr.

冕

䊐

Proof: Immediate. 䊐 If f ⬘ satisfies the Dini condition, we can then define

冕␻

1 log ␭

1

n

⍀ f ⬘共 t 兲 ⫽

851

* 兲 ⫺s 共 ␻ *n 兲兩 ⭐C⍀ 共 ␦ n 兲 , 兩s共 ␻m

共 m⬎n 兲 .

Since ␦ n →0, 兵 s( ␻ n* ) 其 is a Cauchy sequence. The existence of the scaling function s( ␻ ) is proved. Letting m→⬁ in the above inequality, we get 兩 s 共 ␻ * 兲 ⫺s 共 ␻ n* 兲兩 ⭐C⍀ 共 ␦ n 兲 .

Now it is easy to get the claimed result by replacing C by 2C. 䊐 Moreover, we have that Corollary 1: Suppose f is an expanding C 1 folding map whose derivative satisfies the strong Dini condition. Then the scaling function s is of summable variation. Proof: According to Theorem 3,

冕

1

0

⍀ f ⬘共 r 兲

dr ⬍⬁ r

implies that s is of summable variation. Now let us see how to control 兰 10 ⍀ f ⬘ (r) (dr/r) from the strong Dini condition. From


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dr ⫽ lim ⍀ 共 ⑀ 兲 log ⑀ ⫹ r ⑀ →0 f ⬘

⍀ f ⬘共 r 兲

冕␻ 1

0

f ⬘共 r 兲

兩 log r 兩 dr r

n→⬁

and ⍀ f ⬘ 共 ⑀ 兲 log兩 ⑀ 兩 ⭐ we see that

冕␻ 1

0

冕␻ ⑀

0

兰 10 ⍀ f ⬘ (r)

f ⬘共 r 兲

f ⬘共 r 兲

⫽⫺ lim

兩 log r 兩 dr, r

兩 log r 兩 dr⬍⬁. r

if ␻ 苸u j 共 ⌺ 兲 .

n

兿

k⫽0

兩 g ⬘j 共 g j k⫹1 ••• j n 共 x 兲兲兩 ⫺ ␤ . k

Proof: The proof is based on Lemma 1, the naive distortion lemma and the Ruelle transfer operator theorem on symbolic space 共see also Refs. 7,8,10兲. Its details are given in Ref. 11.

III. MULTIFRACTAL ANALYSIS I: THE PROOF OF THEOREM 1

Let x⫽ ␲␻ with ␻ ⫽ j 0 j 1 •••. Let ␻ n ⫽ j 0 ••• j n⫺1 . By Lemma 2, we have 共 f n 兲 ⬘共 x 兲 ⫽

1 g ␻⬘ 共 f n 共 x 兲兲 n

⫽

k

q 共 ␻ 兲 ⫽log兩 g ⬘j 共 ␲␻ 兲兩 0

for ␻ ⫽ j 0 j 1 ••• j n •••. Then we have that 1 n

n

兺

k⫽0

q 共 ␪ k ␻ 兲 ⫽⫺ lim n→⬁

log兩 I ␻ n 兩 n

.

Note that the middle term in the last equation is in the form of ergodic means of the Birkhoff ergodic theorem. Therefore, for any ␪ 共as well as f ) invariant ergodic measure ␯ , L( ␲␻ ) exists and equals a constant for almost every ␯ point. The constant equals ⫺ 兰 ⌺ q( ␻ )d ␯ . Let ␮ ␤ be the Gibbs measure of the operator T ␤ . By the Gibbs property of ␮ ␤ , we have log ␮ ␤ 共 I n 共 x 兲兲 n ⫽⫺ ␤ ⫺ P 共 ␤ 兲 lim , n→⬁ log兩 I n 共 x 兲兩 n→⬁ log兩 I n 共 x 兲兩 lim

where I n (x)⫽I ␻ n means the interval in In containing x. For ␮ ␤ -almost everywhere we also have lim n→⬁

log兩 I ␻ n 共 x 兲兩 n

⫽

冕

⌺

q 共 ␶ 兲 d ␮ ␤ 共 ␶ 兲 ⫽⫺ P ⬘ 共 ␤ 兲

共if P( ␤ ) is differentiable at ␤ ). So, if ␣ ⫽ P ⬘ ( ␤ ), we have ␮ ␤ (E ␣ )⫽1. On the other hand, if x苸E ␣ ,

⭐C

holds for any sequence ␻ n ⫽ j 0 ••• j n⫺1 , any x苸I, where G ␻ n共 x 兲 ⫽

.

Now let us define a function q on ⌺ as

n→⬁

共1兲 The operators S, T 兩 C(I) and T 兩 C(⌳) have the same spectral radius ␳ . 共2兲 There are unique probability measure ␯ and ␮ respectively supported by ⌺ and ⌳ such that S * ␯ ⫽ ␳␯ ,T * ␮ ⫽ ␳ ␮ , where S * and T * are respectively the adjoint operator of S and T. 共3兲 ␮ is the image of ␯ under ␲ . 共4兲 ␮ has the following Gibbs property: there is a constant C⬎0 such that

G ␻ n共 x 兲

k

k

L 共 x 兲 ⫽⫺ lim

Now we see that S ␤ is associated to T ␤ in the following way. For simplicity, we write simply T⫽T ␤ and S⫽S ␤ in the following theorem. Theorem 4: Suppose f is an expanding C 1 folding map with expanding constant ␭⬎1, i.e., 兩 f ⬘ (x) 兩 ⭓␭ and whose derivative satisfies the Dini condition. Then we have

␳

n

k

where

⫺n

log兩 g ⬘j 共 g j k⫹1 ••• j n 共 ␰ 兲兲兩

g ⬘j 共 g j k⫹1 ••• j n 共 ␰ 兲兲 ⫽g ⬘j 共 ␲ ␪ k ␴ 兲 )⫽g ⬘j 共 ␲ 共 j k j k⫹1 ••• 兲兲 .

S ␤ ␸ 共 ␻ 兲 ⫽q ␤ 共 u 0 共 ␻ 兲兲 ␸ 共 u 0 共 ␻ 兲兲 ⫹q ␤ 共 u 1 共 ␻ 兲兲 ␸ 共 u 1 共 ␻ 兲兲 ,

␮共 I ␻n兲

兺

k⫽0

The limit does not depend upon ␰ . So take ␰ ⫽ ␲ ( ␪ n ␻ ). We have

This completes the corollary. 䊐 We have seen that ⌳ is related to ⌺ via ␲ . Define the following transfer operator on C(⌺):

C ⫺1 ⭐

n

log兩 I ␻ n 兩

n→⬁

(dr/r)⬍⬁ if

q ␤ 共 ␻ 兲 ⫽ 兩 g ⬘j 共 ␲␻ 兲兩 ⫺ ␤

1 n

L 共 x 兲 ⫽⫺ lim

1 n⫺1 ⌸ k⫽0 g ⬘j 共 g j k⫹1 ••• j n 共 ␰ 兲兲

log ␮ ␤ 共 I n 共 x 兲兲 P 共 ␤ 兲 ⫺ ␣␤ ⫽⫺ ␤ ⫹ P 共 ␤ 兲 / ␣ ⫽ . ␣ n→⬁ log兩 I n 共 x 兲兩 lim

Now the dimension formula follows from the Billingsley theorem.12 We postpone the proof of differentiability of P( ␤ ) at the end of the proof of Theorem 2.

IV. MULTIFRACTAL ANALYSIS II: THE PROOF OF THEOREM 2

The following result is well-known in symbolic dynamical systems 共see Ref. 13兲. Let ⌽ be a real valued continuous function defined on ⌺ * . Let P ⌽ ( ␤ ) be the pressure of the following transfer operator: T ⌽ ␸ ⫽e ␤ ⌽(u 0* ␻ * ) ␸ 共 u 0* ␻ * 兲 ⫹e ␤ ⌽(u 1* ␻ * ) ␸ 共 u 1* ␻ * 兲 ,

,

k

where ␰ ⫽ f n (x). By the naive distortion lemma, we have

where u i* for i⫽0 and 1 are the inverses of the dual shift ␪ * and can be defined as u i* ( ␻ * )⫽ ␻ * i. If P ⌽ is differentiable ⬘ ( ␤ ), we have at some point ␤ , by noting that ␣ ⫽ P ⌽


Chaos, Vol. 9, No. 4, 1999

再

1 n→⬁ n

dim ␻ * 苸⌺ * : lim ⫽⫺

Exponents and multifractal analysis

n

兺

k⫽0

⌽共 ␪ *k␻ * 兲⫽ ␣

冎

1 关 ␣␤ ⫺ P ⌽ 共 ␤ 兲兴 . log 2

It suffices to apply this to ⌽( ␻ * )⫽⫺log s(␻*). But we have to show that P ⫺log s(␤)⫽P(␤), the pressure function associated with the operators T ␤ , to show that P( ␤ ) is everywhere differentiable. Note that the spectral radius of T ␤ can be expressed as lim n→⬁

1 log n

冉

⫽ lim n→⬁

⫽ lim n→⬁

n⫺1

兺兿

x苸Fix( f n ) k⫽0

兩 f ⬘ 共 f k 共 x 兲兲兩 ␤

冊

1 兩共 f n 兲 ⬘共 x 兲兩 ␤ log n x苸Fix( f n )

兺

1 log n ␻ * 苸Fix( ␪ * n )

兺

n⫺1

兿

k⫽0

兩 s 共 ␪ * k ␻ * 兲兩 ⫺ ␤ ).

共See Ref. 1 pp. 276 and 76 for the above two equalities兲. The last quantity is just the spectral radius of the transfer operator associated with the potential function ⫺ ␤ log s. Thus we have proved the equality P( ␤ )⫽ P ⫺log s(␤). From Corollary 1, the scaling function under the assumption of Theorem 2 is of summable variation. It is well known from Thermodynamical Formalism for symbolic dynamical systems that for a potential being of summable variation, there is a unique equilibrium state corresponding to it.7,10,14 It is also known that the uniqueness in equilibrium states for a given potential function implies the differentiability of the pressure function 共see Ref. 15兲. Therefore, P( ␤ ) and P ⫺log s(␤) are both differentiable at ␤ . Thus we proved Theorem 2 and completed the last argument of the proof of Theorem 1. V. SOME REMARKS

共1兲 Theorem 1 and Theorem 2 provide us the relation, log 2 dim E ␣* . dim E ␣ ⫽ ␣ Note that metrics are different for the two dimensions. The reason for the factor log 2/␣ would be that the shift ␪ has a constant ‘‘Lyapunov exponent’’ log 2 and that the map f restricted on E ␣ has a constant Lyapunov exponent ␣ . 共2兲 If q( ␻ )⫽ 兩 g ⬘j ( ␲␻ ) 兩 is not constant on ⌺, log q(␻) can 0 not be written as u( ␪ ␻ )⫺u( ␻ )⫹C for any continuous function u and any constant C. 16 Then the dimension function in Theorem 2 is strictly concave on some interval.8 共3兲 From a result in Ref. 17, we know that if f is just continuously differentiable, the set L f of all possible

853

Lyapunov exponents is just the set of integrals 兰 log兩 f ⬘兩d␮, where ␮ varies in the set of f -invariant measures.18 共4兲 The method in this paper also works for any C 1 expanding Markov maps satisfying the Dini or Strong Dini condition. See Ref. 1 for Markov maps and their scaling functions.

ACKNOWLEDGMENTS

The final version of this paper was completed when the second author 共Y.J.兲 visited the Faculte´ de Mathe´matiques et Informatique at Universite´ de Picardie Jules Verne in France. During this research the first author 共A.F.兲 visited the Institute of Mathematical Science 共IMS兲 at the Chinese University of Hong Kong and the second author 共Y.J.兲 visited the Institute for Mathematical Research 共FIM兲 at ETH-Zurich in Switzerland. The authors would like to thank these institutes for their hospitality and support. The second author 共Y.J.兲 is supported in part by NSF grants and PSC-CUNY awards.

1

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