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Multifractal analysis of local entropies for recurrence time
Chaos, Solitons and Fractals 33 (2007) 1584–1591 www.elsevier.com/locate/chaos
Multifractal analysis of local entropies for recurrence time Zhenzhen Yan b
a,b,*
, Ercai Chen
q
a
a College of Math and Computer Science, Nanjing Normal University, Nanjing 210097, PR China College of Math and Physics, Nanjing University of Posts and Telecommunications, Nanjing 210003, PR China
Accepted 6 March 2006
Abstract We introduce local entropies and multifractal spectra associated with Poincare´ recurrences. By using characteristics of a dynamical system we establish an exact formula on multifractal spectrum of local entropies for recurrence time. 2006 Elsevier Ltd. All rights reserved.
1. Introduction In 1958 Kolmogorov introduced the concept of entropy into ergodic theory, and this has been the most successful invariant so far. The purpose of computing entropy is to show how dynamics produces chaos. The disorder inherent in a state space can be measured by the entropy of orbits distribution. In a number of physical contexts, very notably turbulence of fluids, so-called ‘‘multifractal scaling’’ is experimentally observed to occur. There are examples in [1] to illustrate two rather common types, the first of multifractal functions and the second of multifractal measures. The term ‘‘multifractal’’ arises from theoretical interpretation of the observed phenomena in terms of a continuous spectrum of ‘‘Ho¨lder singularities’’, usually denoted by h or a according to the context of functions or measures, on sets with ‘‘fractal dimensions’’ ‘‘D(h)’’ or ‘‘f(a)’’, respectively. Historically, multifractal analysis was mainly concerned with the study of pointwise dimensions of a Borel measure l (provided the limit exists): d l ðxÞ ¼ lim e!0
log lðBðx; eÞÞ ; log e
ð1Þ
where B(x, e) is a open e-neighborhood of x. The purpose is to describe the sets of points with a given pointwise dimension. For this the notion of the multifractal spectrum f(a) was introduced in [2]. This spectrum is the function f ðaÞ ¼ dimH ðfx : d l ðxÞ ¼ agÞ;
ð2Þ
where dimH is the Hausdorff dimension. q
This work was partially supported by the National Natural Science Foundation of China (10271057 and 10571086) and the special Funds for Major State Basic Research Projects in China. Z. Yan was partially supported by the Funds for Qing-lan Project of Nanjing University of Posts and Telecommunications. * Corresponding author. Address: College of Math and Physics, Nanjing University of Posts and Telecommunications, Nanjing 210003, PR China. E-mail address: [email protected] (Z. Yan). 0960-0779/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.03.025
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One can generalize this approach even further [3–5]. Physicists are looking at the problem in a very direct way [6–10]. In the instanton approximation to Feynman’s path integral [9,10] the author obtained formula C ¼ expðSÞ; where S is the entropy and ‘‘C’’ is called the ‘‘complexion’’ which is a word borrowed from Latin. However, one can also use following El Naschie [10] S dimH ; where dimH is Hausdorff dimension. So fractals and multifractals come into the subject of entropy in a very direct way. Now we are interested in local entropies and spectra associated with Poincare´ recurrence. These ideas are from [4,5,11– 13]. The dynamical systems considered in this paper will be formed by a compact metric space X with distance d; all the applications f on X will be continuous. The fundamental quantities investigated in this paper are Poincare´ recurrence. Take U a subset of X and define for each x 2 X the first return time into U as sU ðxÞ ¼ inffk > 0jf k ðxÞ 2 U g:
ð3Þ
Define the local entropy for recurrence time at a point x as follows (provided the limit exists): hs ðf ; xÞ ¼ lim lim
e!0 n!1
1 log sBn ðx;eÞ ðxÞ; n
ð4Þ
where Bn(x, e) = {y 2 Xjd(fi(x), fi(y)) < e for i = 0, 1, . . . , n 1}, e > 0. We define the multifractal spectrum for local entropies for recurrence time and the topological entropy of non-compact sets as EðaÞ ¼ htop ðf ; fxjhs ðf ; xÞ ¼ agÞ;
ð5Þ
where htop(f, Z) is the topological entropy of Z. The precise definition is given below,but for the time being one could think it as a dynamical analogue of the Hausdorff dimension. 2. Definitions and lemmas Definition 2.1. We define the lower (resp. upper) local entropies for recurrence time as follows: 1 hs ðf ; xÞ ¼ lim lim inf log sBn ðx;eÞ ðxÞ; e!0 n!1 n hs ðf ; xÞ ¼ lim lim sup 1 log sB ðx;eÞ ðxÞ: n e!0 n n!1
ð6Þ ð7Þ
Note that the limits in e exist by monotonicity. We say that the local entropy for recurrence time exists at x if hs ðf ; xÞ ¼ hs ðf ; xÞ: In this case the common value will be denoted by hs(f, x). The following result establishes the existence of local entropies for recurrence time. Lemma 2.2 (See [14]). Let (X, d) be a compact metric space and f : X ! X be continuous. Let l be an ergodic measure supported on X. For l a.e. x 2 X, one has hs ðf ; xÞ ¼ hs ðf ; xÞ: ð8Þ In this paper we study the multifractal spectrum of local entropies for recurrence time. For every a P 0 consider the level set of local entropies for recurrence time: K a ¼ fx 2 X jhs ðf ; xÞ ¼ ag: Now we give the equivalent definition of topological entropy of non-compact or non-invariant sets which has been introduced by Bowen in [15]. Let U ¼ fU 1 ; U 2 ; . . . ; U M g be a finite open cover of X. A string U is a sequence U i1 U i2 . . . U in with ik 2 {1, 2, S. . . , M}; its length n is denoted by n(U). The collection of all strings of length n is denoted by W n ðUÞ; and W Pn ðUÞ ¼ kPn W k ðUÞ. For each U 2 W n ðUÞ define \ \ \ X ðUÞ ¼ U i1 f 1 U i2 f nþ1 U in ¼ fx 2 X : f k1 2 U ik ; k ¼ 1; 2; . . . ; ng: ð9Þ We say that a collection C of strings covers a set Z X if Z ¨U2CX(U).
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For any real s 2 R and C of strings we introduce the free energy as follows: X expðnðUÞsÞ: F ðC; sÞ ¼ U2C
For a given Z consider the infimum of free energy over all collections C W Pn ðUÞ which cover Z: MðZ; U; s; nÞ ¼
inf
C covers Z
F ðC; sÞ;
ð10Þ
and put MðZ; U; sÞ ¼ lim MðZ; U; s; nÞ: n!1
There exists a unique value sˆ such that MðZ; U; sÞ jump from +1 to 0: hðZ; UÞ ¼ ^s ¼ supfs : MðZ; U; sÞ ¼ þ1g ¼ inffs : MðZ; U; sÞ ¼ 0g:
ð11Þ
Finally, we can show that the following limit exists [4]: htop ðf ; ZÞ ¼
lim
diamðUÞ!0
ð12Þ
hðZ; UÞ:
Definition 2.3. The number htop(f, Z) is called the topological entropy of f restricted to the set Z, or, simply, the topological entropy of Z. This definition of topological entropy is similar to the definition of Hausdorff dimension (the diameters of the covering open sets are replaced by exp(n(U))). Indeed, these definitions are particular cases of the so-called Carathe´odory construction [4] and, therefore, have similar properties. Lemma 2.4 (See [4]). The topological entropy as defined above has the following properties: (i) htop(f, Z1) 6 htop(f, Z2) for any Z1 Z2; (ii) htop(f, ¨ iZi) = supihtop(f, Zi) for any Zi X, i = 1, 2, . . . Following the ideas of [4,5] we introduce the return time-related dimension characteristics hs(f, q, Z), which we call the (q, s)-entropy of f restricted to Z. This definition requires a few steps. Suppose G ¼ fBni ðxi ; eÞgi is any at most countable collection and let t 2 R. We define the (q, t)-free energy of C by X sðBni ðxi ; eÞÞq ðxi Þ expðtni Þ: F s ðG; q; tÞ ¼ i
For any given Z X, Z 5 ;, and number q, t 2 R, e > 0, N 2 N put M cs ðZ; q; t; e; N Þ ¼ inf F s ðG; q; tÞ; G
where S the infimum is taken over all finite or countable collections G ¼ fBni ðxi ; eÞgi with xi 2 Z and ni P N such that Z i Bni ðxi ; eÞ. To complete the definition we put M cs ð;; q; t; e; N Þ ¼ 0; for any q, t, e and N. Since the quantities M cs ðZ; q; t; e; NÞ are non-decreasing in N, the following limit exists: M cs ðZ; q; t; eÞ ¼ lim M cs ðZ; q; t; e; N Þ ¼ sup M cs ðZ; q; t; e; N Þ: N !1
N >1
Since we consider covers with centers in a given set, the quantities M cs ðZ; q; t; eÞ are not necessarily monotonic with respect to the set Z. We enforce monotonicity by putting M s ðZ; q; t; eÞ ¼ sup M cs ðZ 0 ; q; t; eÞ: Z 0 Z
The definition above leads us to prove the following conclusions easily (see [4]). Lemma 2.5. For any t 2 R the set function Ms(Æ, q, t, e) has the following properties: (i) Ms(;, q, t, e) = 0. (ii) Ms(Z s(Z2, q, t, e) for any Z1 Z2. S1, q, t, e) 6 MP (iii) M s ð i Z i ; q; t; eÞ 6 i M s ðZ i ; q; t; eÞ for any Zi X, i = 1, 2, . . .
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Lemma 2.6. There exists a critical value hs(f, q, Z, e) 2 [1, +1] such that 1; t < hs ðf ; q; Z; eÞ; M s ðZ; q; t; eÞ ¼ 0; t > hs ðf ; q; Z; eÞ: Lemma 2.7. The following holds: (i) hs(f, q, ;, e) = 1. (ii) hs(f, q, Z1, e) 6 hs(f, q, Z2, e) for any Z1 Z2. (iii) hs(f, q, ¨ iZi, e) = supi hs(f, q, Zi, e) for any Zi X, i = 1, 2, . . . Now we are interested in the asymptotic behavior of local entropy as e ! 0. Definition 2.8. The (q, s)-entropy of Z is
hs ðf ; q; ZÞ ¼ lim sup hs ðf ; q; Z; eÞ:
ð13Þ
e!0
Let us discuss the existence of the limit with respect to e in the definition above. First, we suppose q 6 0. Let e1 > e2 > 0, and G ¼ fBni ðxi ; e2 Þg is a centered cover of Z. Then obviously G0 ¼ fBni ðxi ; e1 Þg is a cover of Z as well and F s ðG; q; tÞ P F s ðG0 ; q; tÞ: Therefore, Ms(Z, q, t, e2) P Ms(Z, q, t, e1), That is to say hs(f, q, Z, e2) P hs(f, q,Z, e1), and hence the limit in (13) exists. When q > 0, there is no monotonicity with respect to e. However, under an additional assumption we can obtain the monotonic behavior with respect e. So we suppose that for every sufficiently small e > 0 CðeÞ :¼ sup sup n
x
sðBn ðx; 2e ÞÞðxÞ < 1: sðBn ðx; eÞÞðxÞ
ð14Þ
Indeed, we can replace constant 12 by 1a with any a > 1. If formula (14) holds for all e 2 (0, e0), take some e1, e2 2 (0, e0) and e ¼ CðaÞ e let a ¼ ee12 > 1: From formula (14), we can conclude that there exists C < 1; such that s Bn x; ea1 ðxÞ sðBn ðx; e2 ÞÞðxÞ e ¼ < CðaÞ; sðBn ðx; e1 ÞÞðxÞ sðBn ðx; e1 ÞÞðxÞ for all n. Now let G ¼ fBni ðxi ; e2 Þg be a centered cover of Z. Then G0 ¼ fBni ðxi ; e1 Þg is a centered cover of Z as well and e q F s ðG0 ; q; tÞ: Thus M s ðZ; q; t; e2 Þ P C e q M s ðZ; q; t; e1 Þ, and therefore hs(f, q, Z, e2) P hs(f, q, Z, e1). Hence F s ðG; q; tÞ P C the limit in (13) exists. Lemma 2.9. From Lemma 2.7 and Definition 2.8, we can show that hs(f, q, Z) has the following properties: (i) hs(f, q, ;) = 1. (ii) hs(f, q, Z1) 6 hs(f, q, Z2) for any Z1 Z2. (iii) hs(f, q, ¨i Zi) = supihs(f, q, Zi) for any Zi X, i = 1, 2, . . .
3. Main theorem and proof In this section we are going to establish the relation between the topological entropy and (q, s)-entropies of the level sets Ka. Theorem 3.1. For any a P 0 and every q 2 R one has htop ðf ; K a Þ ¼ qa þ hs ðf ; q; K a Þ: Consider a P 0 and the corresponding level set Ka. When x 2 Ka, one has lim lim inf e!0
n!1
1 1 log sðBn ðx; eÞÞðxÞ ¼ lim lim sup log sðBn ðx; eÞÞðxÞ ¼ a: e!0 n n n!1
ð15Þ
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Choose some monotonic sequence eM ! 0 as M ! 1. These sequence will be fixed for the rest of this section. Let d > 0 and put 1 K a;M ¼ x 2 K a ja d < lim inf log sðBn ðx; eM ÞÞðxÞ : n!1 n Obviously, Ka,M Ka,M+1 and Ka = ¨MKa,M. Note that due to the monotonicity of logs(Bn(x, e))(x) in e, for each x 2 Ka and for every e > 0 one has lim sup n!1
1 log sðBn ðx; eÞÞðxÞ < a: n
For fixed x 2 Ka,M there exists N0 = N0(x, d, eM) such that ad<
1 log sðBn ðx; eÞÞðxÞ < a þ d; n
for all n > N0. Put K a;M ;N ¼ fx 2 K a;M jN 0 ¼ N 0 ðx; d; eM Þ < N g:
ð16Þ
Again, it is easy to see that Ka,M,N Ka,M,N+1 and Ka,M = ¨NKa,M,N. Let U be a finite open cover of X. Using the properties of topological entropy we conclude that hðK a ; UÞ ¼ lim lim hðK a;M;N ; UÞ: M !1 N !1
Lemma 3.2. Suppose U is an arbitrary open cover of X. Let dðUÞ is a Lebesgue number of U: Consider Ka,M,N for some M, N 2 N such that eM < dðUÞ 2 : Then for s P qa + jqjd + t one has MðK a;M;N ; U; sÞ 6 M cs ðK a;M;N ; q; t; eM Þ: Proof. Suppose n > N and G ¼ fBni ðxi ; eM Þgi is an arbitrary cover of Ka,M,N with xi 2 Ka,M,N such that ni P n for all i. Since eM < dðUÞ 2 ; for every xi there exists some string U(i) with n(Ui) = ni such that Bni ðxi ; eM Þ X ðUðiÞÞ and hence [ [ Bni ðxi ; eM Þ X ðUi Þ: K a;M ;N i
i
Therefore the collection CG ¼ fUðiÞg of strings covers Ka,M,N. Since xi 2 Ka,M,N for all i and ni P n > N we have expðða dÞni Þ 6 sðBni ðxi ; eM ÞÞðxi Þ 6 expðða þ dÞni Þ: If q P 0, then sðBni ðxi ; eM ÞÞq ðxi Þ > expðqða þ dÞni Þ and X X sðBni ðxi ; eM ÞÞq ðxi Þ expðtni Þ P expðqða þ dÞni Þ expðtni Þ i
i
¼
X
expðni ðqa þ qd þ tÞÞ
i
P
X
expðnðUi ÞsÞ
U 2CG
P MðK a;M;N ; U; s; nÞ; for s P qa + qd + t. Since G is an arbitrary centered covering we get M cs ðK a;M;N ; q; t; eM ; nÞ P MðK a;M;N ; U; s; nÞ; provided s P qa + qd + t. If q < 0, then sðBni ðxi ; eM ÞÞq ðxi Þ > expðqða dÞni Þ and X X sðBni ðxi ; eM ÞÞq ðxi Þ expðtni Þ P expðqða dÞni Þ expðtni Þ i i X expðni ðqa qd þ tÞÞ ¼ i X expðnðUi ÞsÞ P U 2CG
P MðK a;M;N ; U; s; nÞ;
Z. Yan, E. Chen / Chaos, Solitons and Fractals 33 (2007) 1584–1591
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for s P qa qd + t. Again, since G is an arbitrary centered covering we conclude that MðK a;M;N ; U; s; nÞ 6 M cs ðK a;M;N ; q; t; eM ; nÞ: Taking limit as n tends to infinity we obtain the desired result.
h
Theorem 3.3. For any a P 0 and any q 2 R one has htop ðf ; K a Þ 6 qa þ hs ðf ; q; K a Þ: Proof. If Ka = ;, the statement is obvious since both sides are equal to 1. Let us assume that Ka 5 ;. Suppose there exist some a P 0 and some q 2 R such that 1 c ¼ ðhtop ðf ; K a Þ qa hs ðf ; q; K a ÞÞ > 0: 4 Since htop ðf ; K a Þ ¼ lim hðK a ; UÞ; diamðUÞ!0
one can find a finite open cover U such that hðK a ; UÞ > qa þ hs ðf ; q; K a Þ þ 3c: c Let d be an arbitrary positive number if q = 0, and d ¼ 2jqj > 0 if jqj > 0. Consider Ka,M,N defined in (16). Choose sufficiently large M, N such that the following three conditions are satisfied:
hðK a;M;N ; UÞ > qa þ hs ðf ; q; K a Þ þ 2c; 1 eM < dðUÞ; 2 hs ðf ; q; K a Þ þ
ð17Þ
c > hs ðf ; q; K a ; eM Þ: 2
By the definition of hðK a;M;N ; UÞ the inequality in (17) implies MðK a;M;N ; U; qa þ hs ðf ; q; K a Þ þ 2cÞ ¼ þ1: Let s = qa + hs(f, q, Ka) + 2c and t = hs(f, q, Ka) + c jqjd. From Lemma 3.2 we conclude that M s ðK a ; q; hs ðf ; q; K a Þ þ c jqjd; eM Þ ¼ þ1:
ð18Þ
However, hs ðf ; q; K a Þ þ c jqjd ¼ hs ðf ; q; K a Þ þ c
c 2
> hs ðf ; q; K a ; eM Þ P hs ðf ; q; K a;M;N ; eM Þ: By using Lemma 2.6, we conclude M s ðK a ; q; hs ðf ; q; K a Þ þ c jqjd; eM Þ ¼ 0: Therefore we arrive at a contradiction from (18) and (19).
ð19Þ h
Lemma 3.4. Suppose d > 0 and the corresponding set Ka,M,N for some M, N. Let U be any finite open cover with diamðUÞ < 12 eM : Then for any s 6 qa jqjd + t one has M cs ðK a;M;N ; q; t; eM Þ 6 MðK a;M;N ; U; sÞ: Proof. Fixed M, N and let Z Ka,M,N,Z 5 ;. Consider an arbitrary finite open cover U of Z with diamðUÞ < 12 eM : Choose any n > N and let C be an arbitrary collection of strings covering Z with nðCÞ ¼ inf nðUÞ P n > N : U2C
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Without loss of generality we may assume X(U) ˙ Z 5 ; foe any U 2 C. Let x(U) 2 X(U) ˙ Z. Since eM > 2diamðUÞ; one has X ðUÞ BnðUÞ ðxðUÞ; eM Þ: Hence the collection {Bn(U)(x(U), eM)} is a centered cover of Z. Since x(U) 2 Z Ka,M,N and n(U) > N, one has expðnðUÞða dÞÞ 6 sðBnðUÞ ðxðUÞ; eM ÞÞðxðUÞÞ 6 expðnðUÞða þ dÞÞ: If q P 0, then M cs ðZ; q; t; eM ; N Þ 6
X
sðBnðUÞ ðxðUÞ; eM ÞÞq ðxðUÞÞ expðnðUÞtÞ
U 2C
6
X
expðqða dÞÞ expðnðUÞtÞ
U 2C
6
X
6
X
expðnðUÞðqa qd þ tÞÞ
U2C
expðnðUÞsÞ;
U2C
for s 6 qa qd + t. If q < 0, then M cs ðZ; q; t; eM ; N Þ 6
X
sðBnðUÞ ðxðUÞ; eM ÞÞq ðxðUÞÞ expðnðUÞtÞ
U 2C
6
X
expðqða þ dÞÞ expðnðUÞtÞ
U 2C
6
X
6
X
expðnðUÞðqa þ qd þ tÞÞ
U2C
expðnðUÞsÞ;
U2C
for s 6 a + qd + t. Since C is an arbitrary covering of Z by strings of length at least n, for s 6 qa jqjd + t, we get M cs ðZ; q; t; eM ; nÞ 6 MðZ; U; s; nÞ: Therefore, M cs ðZ; q; t; eM Þ 6 MðZ; U; sÞ 6 MðK a;M;N ; U; sÞ:
Theorem 3.5. For any a P 0 and any q 2 R one has htop ðf ; K a Þ P qa þ hs ðf ; q; K a Þ: Proof. Using arguments similar to the proof of Theorem 3.3, we can prove the result.
h
Proof of Theorem 3.1. Combining the results of Theorems 3.3 and 3.5, we easily obtain the conclusion.
h
Acknowledgement We would like to thank Prof. M.S. El Naschie for valuable suggestions.
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Multifractal analysis of local entropies for recurrence time Zhenzhen Yan b
a,b,*
, Ercai Chen
q
a
a College of Math and Computer Science, Nanjing Normal University, Nanjing 210097, PR China College of Math and Physics, Nanjing University of Posts and Telecommunications, Nanjing 210003, PR China
Accepted 6 March 2006
Abstract We introduce local entropies and multifractal spectra associated with Poincare´ recurrences. By using characteristics of a dynamical system we establish an exact formula on multifractal spectrum of local entropies for recurrence time. 2006 Elsevier Ltd. All rights reserved.
1. Introduction In 1958 Kolmogorov introduced the concept of entropy into ergodic theory, and this has been the most successful invariant so far. The purpose of computing entropy is to show how dynamics produces chaos. The disorder inherent in a state space can be measured by the entropy of orbits distribution. In a number of physical contexts, very notably turbulence of fluids, so-called ‘‘multifractal scaling’’ is experimentally observed to occur. There are examples in [1] to illustrate two rather common types, the first of multifractal functions and the second of multifractal measures. The term ‘‘multifractal’’ arises from theoretical interpretation of the observed phenomena in terms of a continuous spectrum of ‘‘Ho¨lder singularities’’, usually denoted by h or a according to the context of functions or measures, on sets with ‘‘fractal dimensions’’ ‘‘D(h)’’ or ‘‘f(a)’’, respectively. Historically, multifractal analysis was mainly concerned with the study of pointwise dimensions of a Borel measure l (provided the limit exists): d l ðxÞ ¼ lim e!0
log lðBðx; eÞÞ ; log e
ð1Þ
where B(x, e) is a open e-neighborhood of x. The purpose is to describe the sets of points with a given pointwise dimension. For this the notion of the multifractal spectrum f(a) was introduced in [2]. This spectrum is the function f ðaÞ ¼ dimH ðfx : d l ðxÞ ¼ agÞ;
ð2Þ
where dimH is the Hausdorff dimension. q
This work was partially supported by the National Natural Science Foundation of China (10271057 and 10571086) and the special Funds for Major State Basic Research Projects in China. Z. Yan was partially supported by the Funds for Qing-lan Project of Nanjing University of Posts and Telecommunications. * Corresponding author. Address: College of Math and Physics, Nanjing University of Posts and Telecommunications, Nanjing 210003, PR China. E-mail address: [email protected] (Z. Yan). 0960-0779/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.03.025
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One can generalize this approach even further [3–5]. Physicists are looking at the problem in a very direct way [6–10]. In the instanton approximation to Feynman’s path integral [9,10] the author obtained formula C ¼ expðSÞ; where S is the entropy and ‘‘C’’ is called the ‘‘complexion’’ which is a word borrowed from Latin. However, one can also use following El Naschie [10] S dimH ; where dimH is Hausdorff dimension. So fractals and multifractals come into the subject of entropy in a very direct way. Now we are interested in local entropies and spectra associated with Poincare´ recurrence. These ideas are from [4,5,11– 13]. The dynamical systems considered in this paper will be formed by a compact metric space X with distance d; all the applications f on X will be continuous. The fundamental quantities investigated in this paper are Poincare´ recurrence. Take U a subset of X and define for each x 2 X the first return time into U as sU ðxÞ ¼ inffk > 0jf k ðxÞ 2 U g:
ð3Þ
Define the local entropy for recurrence time at a point x as follows (provided the limit exists): hs ðf ; xÞ ¼ lim lim
e!0 n!1
1 log sBn ðx;eÞ ðxÞ; n
ð4Þ
where Bn(x, e) = {y 2 Xjd(fi(x), fi(y)) < e for i = 0, 1, . . . , n 1}, e > 0. We define the multifractal spectrum for local entropies for recurrence time and the topological entropy of non-compact sets as EðaÞ ¼ htop ðf ; fxjhs ðf ; xÞ ¼ agÞ;
ð5Þ
where htop(f, Z) is the topological entropy of Z. The precise definition is given below,but for the time being one could think it as a dynamical analogue of the Hausdorff dimension. 2. Definitions and lemmas Definition 2.1. We define the lower (resp. upper) local entropies for recurrence time as follows: 1 hs ðf ; xÞ ¼ lim lim inf log sBn ðx;eÞ ðxÞ; e!0 n!1 n hs ðf ; xÞ ¼ lim lim sup 1 log sB ðx;eÞ ðxÞ: n e!0 n n!1
ð6Þ ð7Þ
Note that the limits in e exist by monotonicity. We say that the local entropy for recurrence time exists at x if hs ðf ; xÞ ¼ hs ðf ; xÞ: In this case the common value will be denoted by hs(f, x). The following result establishes the existence of local entropies for recurrence time. Lemma 2.2 (See [14]). Let (X, d) be a compact metric space and f : X ! X be continuous. Let l be an ergodic measure supported on X. For l a.e. x 2 X, one has hs ðf ; xÞ ¼ hs ðf ; xÞ: ð8Þ In this paper we study the multifractal spectrum of local entropies for recurrence time. For every a P 0 consider the level set of local entropies for recurrence time: K a ¼ fx 2 X jhs ðf ; xÞ ¼ ag: Now we give the equivalent definition of topological entropy of non-compact or non-invariant sets which has been introduced by Bowen in [15]. Let U ¼ fU 1 ; U 2 ; . . . ; U M g be a finite open cover of X. A string U is a sequence U i1 U i2 . . . U in with ik 2 {1, 2, S. . . , M}; its length n is denoted by n(U). The collection of all strings of length n is denoted by W n ðUÞ; and W Pn ðUÞ ¼ kPn W k ðUÞ. For each U 2 W n ðUÞ define \ \ \ X ðUÞ ¼ U i1 f 1 U i2 f nþ1 U in ¼ fx 2 X : f k1 2 U ik ; k ¼ 1; 2; . . . ; ng: ð9Þ We say that a collection C of strings covers a set Z X if Z ¨U2CX(U).
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For any real s 2 R and C of strings we introduce the free energy as follows: X expðnðUÞsÞ: F ðC; sÞ ¼ U2C
For a given Z consider the infimum of free energy over all collections C W Pn ðUÞ which cover Z: MðZ; U; s; nÞ ¼
inf
C covers Z
F ðC; sÞ;
ð10Þ
and put MðZ; U; sÞ ¼ lim MðZ; U; s; nÞ: n!1
There exists a unique value sˆ such that MðZ; U; sÞ jump from +1 to 0: hðZ; UÞ ¼ ^s ¼ supfs : MðZ; U; sÞ ¼ þ1g ¼ inffs : MðZ; U; sÞ ¼ 0g:
ð11Þ
Finally, we can show that the following limit exists [4]: htop ðf ; ZÞ ¼
lim
diamðUÞ!0
ð12Þ
hðZ; UÞ:
Definition 2.3. The number htop(f, Z) is called the topological entropy of f restricted to the set Z, or, simply, the topological entropy of Z. This definition of topological entropy is similar to the definition of Hausdorff dimension (the diameters of the covering open sets are replaced by exp(n(U))). Indeed, these definitions are particular cases of the so-called Carathe´odory construction [4] and, therefore, have similar properties. Lemma 2.4 (See [4]). The topological entropy as defined above has the following properties: (i) htop(f, Z1) 6 htop(f, Z2) for any Z1 Z2; (ii) htop(f, ¨ iZi) = supihtop(f, Zi) for any Zi X, i = 1, 2, . . . Following the ideas of [4,5] we introduce the return time-related dimension characteristics hs(f, q, Z), which we call the (q, s)-entropy of f restricted to Z. This definition requires a few steps. Suppose G ¼ fBni ðxi ; eÞgi is any at most countable collection and let t 2 R. We define the (q, t)-free energy of C by X sðBni ðxi ; eÞÞq ðxi Þ expðtni Þ: F s ðG; q; tÞ ¼ i
For any given Z X, Z 5 ;, and number q, t 2 R, e > 0, N 2 N put M cs ðZ; q; t; e; N Þ ¼ inf F s ðG; q; tÞ; G
where S the infimum is taken over all finite or countable collections G ¼ fBni ðxi ; eÞgi with xi 2 Z and ni P N such that Z i Bni ðxi ; eÞ. To complete the definition we put M cs ð;; q; t; e; N Þ ¼ 0; for any q, t, e and N. Since the quantities M cs ðZ; q; t; e; NÞ are non-decreasing in N, the following limit exists: M cs ðZ; q; t; eÞ ¼ lim M cs ðZ; q; t; e; N Þ ¼ sup M cs ðZ; q; t; e; N Þ: N !1
N >1
Since we consider covers with centers in a given set, the quantities M cs ðZ; q; t; eÞ are not necessarily monotonic with respect to the set Z. We enforce monotonicity by putting M s ðZ; q; t; eÞ ¼ sup M cs ðZ 0 ; q; t; eÞ: Z 0 Z
The definition above leads us to prove the following conclusions easily (see [4]). Lemma 2.5. For any t 2 R the set function Ms(Æ, q, t, e) has the following properties: (i) Ms(;, q, t, e) = 0. (ii) Ms(Z s(Z2, q, t, e) for any Z1 Z2. S1, q, t, e) 6 MP (iii) M s ð i Z i ; q; t; eÞ 6 i M s ðZ i ; q; t; eÞ for any Zi X, i = 1, 2, . . .
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Lemma 2.6. There exists a critical value hs(f, q, Z, e) 2 [1, +1] such that 1; t < hs ðf ; q; Z; eÞ; M s ðZ; q; t; eÞ ¼ 0; t > hs ðf ; q; Z; eÞ: Lemma 2.7. The following holds: (i) hs(f, q, ;, e) = 1. (ii) hs(f, q, Z1, e) 6 hs(f, q, Z2, e) for any Z1 Z2. (iii) hs(f, q, ¨ iZi, e) = supi hs(f, q, Zi, e) for any Zi X, i = 1, 2, . . . Now we are interested in the asymptotic behavior of local entropy as e ! 0. Definition 2.8. The (q, s)-entropy of Z is
hs ðf ; q; ZÞ ¼ lim sup hs ðf ; q; Z; eÞ:
ð13Þ
e!0
Let us discuss the existence of the limit with respect to e in the definition above. First, we suppose q 6 0. Let e1 > e2 > 0, and G ¼ fBni ðxi ; e2 Þg is a centered cover of Z. Then obviously G0 ¼ fBni ðxi ; e1 Þg is a cover of Z as well and F s ðG; q; tÞ P F s ðG0 ; q; tÞ: Therefore, Ms(Z, q, t, e2) P Ms(Z, q, t, e1), That is to say hs(f, q, Z, e2) P hs(f, q,Z, e1), and hence the limit in (13) exists. When q > 0, there is no monotonicity with respect to e. However, under an additional assumption we can obtain the monotonic behavior with respect e. So we suppose that for every sufficiently small e > 0 CðeÞ :¼ sup sup n
x
sðBn ðx; 2e ÞÞðxÞ < 1: sðBn ðx; eÞÞðxÞ
ð14Þ
Indeed, we can replace constant 12 by 1a with any a > 1. If formula (14) holds for all e 2 (0, e0), take some e1, e2 2 (0, e0) and e ¼ CðaÞ e let a ¼ ee12 > 1: From formula (14), we can conclude that there exists C < 1; such that s Bn x; ea1 ðxÞ sðBn ðx; e2 ÞÞðxÞ e ¼ < CðaÞ; sðBn ðx; e1 ÞÞðxÞ sðBn ðx; e1 ÞÞðxÞ for all n. Now let G ¼ fBni ðxi ; e2 Þg be a centered cover of Z. Then G0 ¼ fBni ðxi ; e1 Þg is a centered cover of Z as well and e q F s ðG0 ; q; tÞ: Thus M s ðZ; q; t; e2 Þ P C e q M s ðZ; q; t; e1 Þ, and therefore hs(f, q, Z, e2) P hs(f, q, Z, e1). Hence F s ðG; q; tÞ P C the limit in (13) exists. Lemma 2.9. From Lemma 2.7 and Definition 2.8, we can show that hs(f, q, Z) has the following properties: (i) hs(f, q, ;) = 1. (ii) hs(f, q, Z1) 6 hs(f, q, Z2) for any Z1 Z2. (iii) hs(f, q, ¨i Zi) = supihs(f, q, Zi) for any Zi X, i = 1, 2, . . .
3. Main theorem and proof In this section we are going to establish the relation between the topological entropy and (q, s)-entropies of the level sets Ka. Theorem 3.1. For any a P 0 and every q 2 R one has htop ðf ; K a Þ ¼ qa þ hs ðf ; q; K a Þ: Consider a P 0 and the corresponding level set Ka. When x 2 Ka, one has lim lim inf e!0
n!1
1 1 log sðBn ðx; eÞÞðxÞ ¼ lim lim sup log sðBn ðx; eÞÞðxÞ ¼ a: e!0 n n n!1
ð15Þ
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Choose some monotonic sequence eM ! 0 as M ! 1. These sequence will be fixed for the rest of this section. Let d > 0 and put 1 K a;M ¼ x 2 K a ja d < lim inf log sðBn ðx; eM ÞÞðxÞ : n!1 n Obviously, Ka,M Ka,M+1 and Ka = ¨MKa,M. Note that due to the monotonicity of logs(Bn(x, e))(x) in e, for each x 2 Ka and for every e > 0 one has lim sup n!1
1 log sðBn ðx; eÞÞðxÞ < a: n
For fixed x 2 Ka,M there exists N0 = N0(x, d, eM) such that ad<
1 log sðBn ðx; eÞÞðxÞ < a þ d; n
for all n > N0. Put K a;M ;N ¼ fx 2 K a;M jN 0 ¼ N 0 ðx; d; eM Þ < N g:
ð16Þ
Again, it is easy to see that Ka,M,N Ka,M,N+1 and Ka,M = ¨NKa,M,N. Let U be a finite open cover of X. Using the properties of topological entropy we conclude that hðK a ; UÞ ¼ lim lim hðK a;M;N ; UÞ: M !1 N !1
Lemma 3.2. Suppose U is an arbitrary open cover of X. Let dðUÞ is a Lebesgue number of U: Consider Ka,M,N for some M, N 2 N such that eM < dðUÞ 2 : Then for s P qa + jqjd + t one has MðK a;M;N ; U; sÞ 6 M cs ðK a;M;N ; q; t; eM Þ: Proof. Suppose n > N and G ¼ fBni ðxi ; eM Þgi is an arbitrary cover of Ka,M,N with xi 2 Ka,M,N such that ni P n for all i. Since eM < dðUÞ 2 ; for every xi there exists some string U(i) with n(Ui) = ni such that Bni ðxi ; eM Þ X ðUðiÞÞ and hence [ [ Bni ðxi ; eM Þ X ðUi Þ: K a;M ;N i
i
Therefore the collection CG ¼ fUðiÞg of strings covers Ka,M,N. Since xi 2 Ka,M,N for all i and ni P n > N we have expðða dÞni Þ 6 sðBni ðxi ; eM ÞÞðxi Þ 6 expðða þ dÞni Þ: If q P 0, then sðBni ðxi ; eM ÞÞq ðxi Þ > expðqða þ dÞni Þ and X X sðBni ðxi ; eM ÞÞq ðxi Þ expðtni Þ P expðqða þ dÞni Þ expðtni Þ i
i
¼
X
expðni ðqa þ qd þ tÞÞ
i
P
X
expðnðUi ÞsÞ
U 2CG
P MðK a;M;N ; U; s; nÞ; for s P qa + qd + t. Since G is an arbitrary centered covering we get M cs ðK a;M;N ; q; t; eM ; nÞ P MðK a;M;N ; U; s; nÞ; provided s P qa + qd + t. If q < 0, then sðBni ðxi ; eM ÞÞq ðxi Þ > expðqða dÞni Þ and X X sðBni ðxi ; eM ÞÞq ðxi Þ expðtni Þ P expðqða dÞni Þ expðtni Þ i i X expðni ðqa qd þ tÞÞ ¼ i X expðnðUi ÞsÞ P U 2CG
P MðK a;M;N ; U; s; nÞ;
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for s P qa qd + t. Again, since G is an arbitrary centered covering we conclude that MðK a;M;N ; U; s; nÞ 6 M cs ðK a;M;N ; q; t; eM ; nÞ: Taking limit as n tends to infinity we obtain the desired result.
h
Theorem 3.3. For any a P 0 and any q 2 R one has htop ðf ; K a Þ 6 qa þ hs ðf ; q; K a Þ: Proof. If Ka = ;, the statement is obvious since both sides are equal to 1. Let us assume that Ka 5 ;. Suppose there exist some a P 0 and some q 2 R such that 1 c ¼ ðhtop ðf ; K a Þ qa hs ðf ; q; K a ÞÞ > 0: 4 Since htop ðf ; K a Þ ¼ lim hðK a ; UÞ; diamðUÞ!0
one can find a finite open cover U such that hðK a ; UÞ > qa þ hs ðf ; q; K a Þ þ 3c: c Let d be an arbitrary positive number if q = 0, and d ¼ 2jqj > 0 if jqj > 0. Consider Ka,M,N defined in (16). Choose sufficiently large M, N such that the following three conditions are satisfied:
hðK a;M;N ; UÞ > qa þ hs ðf ; q; K a Þ þ 2c; 1 eM < dðUÞ; 2 hs ðf ; q; K a Þ þ
ð17Þ
c > hs ðf ; q; K a ; eM Þ: 2
By the definition of hðK a;M;N ; UÞ the inequality in (17) implies MðK a;M;N ; U; qa þ hs ðf ; q; K a Þ þ 2cÞ ¼ þ1: Let s = qa + hs(f, q, Ka) + 2c and t = hs(f, q, Ka) + c jqjd. From Lemma 3.2 we conclude that M s ðK a ; q; hs ðf ; q; K a Þ þ c jqjd; eM Þ ¼ þ1:
ð18Þ
However, hs ðf ; q; K a Þ þ c jqjd ¼ hs ðf ; q; K a Þ þ c
c 2
> hs ðf ; q; K a ; eM Þ P hs ðf ; q; K a;M;N ; eM Þ: By using Lemma 2.6, we conclude M s ðK a ; q; hs ðf ; q; K a Þ þ c jqjd; eM Þ ¼ 0: Therefore we arrive at a contradiction from (18) and (19).
ð19Þ h
Lemma 3.4. Suppose d > 0 and the corresponding set Ka,M,N for some M, N. Let U be any finite open cover with diamðUÞ < 12 eM : Then for any s 6 qa jqjd + t one has M cs ðK a;M;N ; q; t; eM Þ 6 MðK a;M;N ; U; sÞ: Proof. Fixed M, N and let Z Ka,M,N,Z 5 ;. Consider an arbitrary finite open cover U of Z with diamðUÞ < 12 eM : Choose any n > N and let C be an arbitrary collection of strings covering Z with nðCÞ ¼ inf nðUÞ P n > N : U2C
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Without loss of generality we may assume X(U) ˙ Z 5 ; foe any U 2 C. Let x(U) 2 X(U) ˙ Z. Since eM > 2diamðUÞ; one has X ðUÞ BnðUÞ ðxðUÞ; eM Þ: Hence the collection {Bn(U)(x(U), eM)} is a centered cover of Z. Since x(U) 2 Z Ka,M,N and n(U) > N, one has expðnðUÞða dÞÞ 6 sðBnðUÞ ðxðUÞ; eM ÞÞðxðUÞÞ 6 expðnðUÞða þ dÞÞ: If q P 0, then M cs ðZ; q; t; eM ; N Þ 6
X
sðBnðUÞ ðxðUÞ; eM ÞÞq ðxðUÞÞ expðnðUÞtÞ
U 2C
6
X
expðqða dÞÞ expðnðUÞtÞ
U 2C
6
X
6
X
expðnðUÞðqa qd þ tÞÞ
U2C
expðnðUÞsÞ;
U2C
for s 6 qa qd + t. If q < 0, then M cs ðZ; q; t; eM ; N Þ 6
X
sðBnðUÞ ðxðUÞ; eM ÞÞq ðxðUÞÞ expðnðUÞtÞ
U 2C
6
X
expðqða þ dÞÞ expðnðUÞtÞ
U 2C
6
X
6
X
expðnðUÞðqa þ qd þ tÞÞ
U2C
expðnðUÞsÞ;
U2C
for s 6 a + qd + t. Since C is an arbitrary covering of Z by strings of length at least n, for s 6 qa jqjd + t, we get M cs ðZ; q; t; eM ; nÞ 6 MðZ; U; s; nÞ: Therefore, M cs ðZ; q; t; eM Þ 6 MðZ; U; sÞ 6 MðK a;M;N ; U; sÞ:
Theorem 3.5. For any a P 0 and any q 2 R one has htop ðf ; K a Þ P qa þ hs ðf ; q; K a Þ: Proof. Using arguments similar to the proof of Theorem 3.3, we can prove the result.
h
Proof of Theorem 3.1. Combining the results of Theorems 3.3 and 3.5, we easily obtain the conclusion.
h
Acknowledgement We would like to thank Prof. M.S. El Naschie for valuable suggestions.
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