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Variational analysis for the multifractal spectra of local entropies and Lyapunov exponents
Chaos, Solitons and Fractals 19 (2004) 1031–1038 www.elsevier.com/locate/chaos
Variational analysis for the multifractal spectra of local entropies and Lyapunov exponents n *, Fernando Vericat Alejandro M. Meso Instituto de Fısica de Lıquidos y Sistemas Biol ogicos (IFLYSIB), UNLP-CONICET, cc. 565, 1990 La Plata, Argentina Grupo de Aplicaciones Matem aticas y Estadısticas de la Facultad de Ingenierıa (GAMEFI), Departamento de Fisicomatem atica, Facultad de Ingenierıa, Universidad Nacional de La Plata, Argentina Accepted 22 April 2003 Communicated by I. Procaccia
Abstract In a previous article [Chaos, Solitons and Fractals, 13 (2002) 1037], the authors have analyzed the multifractal Lyapunov spectrum. Here we continue that study by considering perturbations of the potential and the dynamics to obtain variational expressions for the entropies and Lyapunov spectra. The spirit and the framework of this note is to obtain, beyond hyperbolicity, variational results, some of which are new and some other which have already been derived but under stronger conditions. 2003 Elsevier Ltd. All rights reserved.
1. Introduction The concept of ‘‘multifractal analysis’’ was raised in the physical ambient [1,2]. One of the main motivations was the study of the behavior of physical measures on strange attractors, as well as aspects related with diffusion-limited, aggregates, etc. [9]. Among another areas of interest it could be mentioned: the study of turbulence [11], distribution and cluster of galaxies [13] and some subjects in Biology. In the study of chaotic behaviors are frequently found invariant sets with a complex mathematical structure. The multifractal analysis essentially treats with decompositions, the so-called fractal decomposition, of these sort of sets. Despite the complexity of the level sets, in many cases it can be found a well comported function which encodes the information about the decomposition. Let consider for instance a decomposition of an attractor A, carrying an invariant measure l, in sets Ka ¼ f x : lðBr ðxÞÞ ra as r ! 0g; ð1Þ where Br ðxÞ is the ball of centre x and radius r. The function fa ðxÞ ¼ dimH Ka , the Haussdorf dimension of the set Ka , was studied in [8,10], these approaches were generalized to study local (depending on points) characteristics of dynamical systems. A complete description of the multifractal analysis of invariant measures was performed by Pesin and Weiss in [15]. In this work were extended all the results known until that moment about smooth conformal maps. The general concept of multifractal analysis was introduced in [3], we present here a brief account: Let X be a set and g : X ! ½1; þ1, now the sets in which X is partitioned are Ka ¼ Ka ðgÞ ¼ fx : gðxÞ ¼ ag:
ð2Þ
* Corresponding author. Address: Instituto de Fısica de Lıquidos y Sistemas Biol ogicos (IFLYSIB), UNLP-CONICET, cc. 565, 1990 La Plata, Argentina. Fax: +54-221-425-7317. E-mail addresses: meson@iflysib.unlp.edu.ar (A.M. Mes on), vericat@iflysib.unlp.edu.ar (F. Vericat).
0960-0779/$ - see front matter 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0960-0779(03)00204-2
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Let G be a function defined on sets, and let F ðaÞ ¼ GðKa Þ, the map F is called the multifractal spectrum specified by the pair ðg; GÞ. Examples of such multifractal spectra are gðxÞ ¼ Dl ðxÞ, the pointwise dimension of the measure l, gðxÞ ¼ hðf ; xÞ, the local entropy or gðxÞ ¼ kðxÞ, Lyapunov exponents. In the paper by Barreira et al. [3], a complete multifractal analysis for pointwise dimensions, local entropies as well as for Lyapunov exponents of conformal expanding maps and Axiom A-diffeomorphisms was performed. Accordingly, for these dynamical systems, the existence of Markov partitions and Gibbs measures is ensured. Results of that article, concerning to local pointwise entropies, were recovered, under weaker hypothesis, by Takens and Verbitski in [17]. In [14] we gave a description of the Lyapunov spectrum, under hyphothesis like Takens–Verbitski. In this article we study the variational properties of perturbations on the entropies and Lyapunov spectra. More specifically we shall consider firstly, for a fixed dynamical map f : X ! X a family of perturbative potentials with respect to u0 : U ¼ fuk gk2ðd;dÞ and study the variation of the entropies spectrum, by analyzing the derivatives of function sðk; qÞ which is in turn is a perturbation of the function T ðqÞ ¼ P ðqu0 Þ qP ðu0 Þ (P is the topological pressure) whose Legendre transformation describes the spectrum of local entropies for the non-perturbative case [17]. For the Lyapunov spectrum we consider more general perturbations, besides a family of potentials we introduce a family of dynamics F ¼ ffk gk2ðd;dÞ . According to [14], the perturbative functions to be considered must be sðk; qÞ, which is determined from P ðqhtop ðfk Þ sðk; qÞwðkÞÞ ¼ 0;
ð3Þ
with wðk; xÞ :¼ log kDx ðfk Þk. Here the topological pressure is that associated to uk and fk . The study of the effect of these perturbations is very important for numerical computations, the numerical results could be affected by small perturbations and it may be interesting to know the influence of the perturbations. Variational results about dimension spectra were obtained in [4,19] for hyperbolic diffeomorphisms. In this article we shall derive variational formula, but under much weaker hyphothesis. The paper is organized as follows. In next section we recall some definitions and relevant results which will be useful for further demonstrations. There we also declare our hypothesis. In Section 3 we announce and prove our results.
2. Previous definitions Let M be a compact metric space, and f : M ! M a continuous map. A function u : M ! R, will be called a ‘‘potential’’. The topological pressure associated to a given potential u is the number [12–18]: Z P ðuÞ ¼ sup hl ðf Þ þ u dl ; ð4Þ l
where the supreme is taken over all the Borel measures l, f ––invariant on M, and where hl ðf Þ is the usual Kolmogorov measure-theoretic entropy of f . An equilibrium state for the potential u is a measure lu which satisfies: Z ð5Þ P ðuÞ ¼ hlu ðf Þ þ u dlu : Under certain conditions imposed on the map f and the potential u an equilibrium state can be constructed [12–17]. The specification property for a map f : M ! M intuitively says that, for specified trajectory segments, a periodic orbit that approximates them can be found. This condition ensures abundance of periodic points. The formal definition of this concept, as it was introduced by Bowen in [5], reads: A homeomorphism f : M ! M have the specification property if given a finite disjoint collection of integer intervals I1 ; . . . ; Ik then, for e > 0, there is a integer MðeÞ and a function U : I ¼ [Ii ! M such that (i) distðIi ; Ij Þ > MðeÞ (Euclidean distance) (ii) f n1 n2 ðUðn1 ÞÞ ¼ Uðn2 Þ (iii) dðf n ðxÞ; UðnÞÞ < e, for some x : f m ðxÞ ¼ x, with m P MðeÞ þ lengthðIÞ and for every n 2 I. A homeomorphism f : M ! M is called expansive if there is a constant d > 0, such that dðf n ðxÞ; f n ðyÞÞ < d, for any integer n implies x ¼ y. Notice that both definitions (about specification and expansiveness) were done without using a differentiable structure. They are topological concepts.
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The following metric in M, is derived from the original one d: dn ðx; yÞ ¼ max fdðf i ðxÞ; f i ðyÞÞ : i ¼ 0; 1; . . . ; n 1g:
ð6Þ
The ball of centre x and radius e in the metric dn will be denoted by Bn;e ðxÞ. For a potential u we put Sn ðuÞðxÞ ¼
n1 X
uðf i ðxÞÞ:
ð7Þ
i¼0
Following [12] or [17], we say that a potential u belongs to the class mf ðMÞ if this condition is fulfilled: There are constants e, K > 0 in a such way that dn ðx; yÞ < e ) jSn ðuÞðxÞ Sn ðuÞðyÞj < K:
ð8Þ
Now we recall how a measure can be defined in such a way that it results to be an equilibrium state: Let Pn ðf Þ ¼ fx : f n ðxÞ ¼ xg, set for a potential u: lu;n ¼
X 1 expðSn ðuÞðxÞÞdx ; N ðf ; u; nÞ x2P ðf Þ
ð9Þ
n
P where N ðf ; u; nÞ ¼ x2Pn ðf Þ expðSn ðuÞðxÞÞ and dx is the unit measure at x. The weak limit of the sequence flu;n g, is an equilibrium state lu , i.e. Z Z gðxÞ dlu;n ¼ gðxÞ dlu ; lim n!1
ð10Þ
for every continuous gðxÞ [12–16]. A measure l on M is called a Gibbs state if, for sufficiently small e > 0, there are constants Ae , Be > 0, such that, for any x 2 M and for any positive integer n, is Ae ðexpðSn ðuÞðxÞÞ nP ðuÞÞ 6 lðBn;e ðxÞÞ 6 Be ðexpðSn ðuÞðxÞÞ nP ðuÞÞ:
ð11Þ
Theorem 1 [12–16]. Let f be an expansive homeomorphism which have the specification property and u a potential belonging to the class mf ðMÞ, then lu is an equilibrium state associated to u, which is a Gibbs state. Besides it is ergodic. We recall the definition of Lyapunov exponents. Let M be a compact Riemannian differentiable manifold d–– dimensional. Let f : M ! M be a C 2 diffeomorphism; we denote by Lx the differential map of f in x, and by Lnx the differential map of f n in x. If v is a tangent vector to M in x, let vðxÞ ¼ vðx; v; f Þ :¼ lim
n!1
1 log Lnx v: n
ð12Þ
The number vðxÞ ¼ vðx; v; f Þ is called the Lyapunov exponent with respect to L ¼ Lðf Þ in ðx; vÞ, provided the limit exists. For a real number v and x 2 M, we denote Ev ðxÞ ¼ fv : vðx; v; f Þ 6 vg: Notice that if v1 6 v2 then Ev1 ðxÞ Ev2 ðxÞ. Furthermore, if x 2 M then there exists an integer mðxÞ 6 d, such that there is a collection of numbers v1 ðxÞ; v2 ðxÞ; . . . ; vmðxÞ ðxÞ and linear subspaces Ev1 ðxÞ; Ev2 ðxÞ; . . . ; EvmðxÞ ðxÞ, with k1 ðxÞ < k2 ðxÞ < < kmðxÞ ðxÞ; Ek1 ðxÞ Ek2 ðxÞ EkmðxÞ ðxÞ: If v 2 Eviþ1 ðxÞ Evi ðxÞ then vðx; vÞ ¼ viþ1 ðxÞ. The Lyapunov spectrum is now defined as Spx ðL; f Þ ¼ fvi ðxÞ : i ¼ 1; 2; . . . ; mðxÞg:
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Our main hypothesis will be (a) For the entropies spectrum f : M ! M, with M a compact metric space, an expansive homeomorphism with specification property. (b) For the Lyapunov spectrum • f : M ! M a C 2 expansive diffeomorphism and with specification property, M a compact Riemannian manifold. • The differential map of f satisfies: there are constants e, K > 0 in a such way that dn ðx; yÞ < e ) Lnx Lny < K; so that the potential wðxÞ ¼ log kLx k belongs to the class mf ðMÞ. Indeed the statistical sum Sn ðwÞðxÞ is equal to n1 X
n1 Y Li ¼ log Ln : log Lix ¼ log x x
i¼0
i¼0
To finish this section we recall the following definition: let f : M ! M be a continuous map and M a compact metric space. The upper and lower local entropies are hl ðx; f Þ ¼ lim lim sup e!0 n!1
hl ðx; f Þ ¼ lim lim inf e!0 n!1
1 log lðBe;n ðxÞÞ; n
1 log lðBe;n ðxÞÞ: n
Then (Brin–Katok theorem [7]), the local entropy does exist, i.e. hl ðx; f Þ ¼ hl ðx; f Þ :¼ hl ðx; f Þ, for every x 2 X .
3. Results 3.1. Entropies spectrum We begin by recalling the main facts of the multifractal formalism developed by Takens and Verbitski [17]. ii(i) The function T ðqÞ ¼ Tu ðqÞ :¼ P ðquÞ qP ðuÞ, u 2 mf ðMÞ, is convex and continuously differentiable with T 0 ðqÞ ¼ R u dlq , where lq is an equilibrium state for qu. This map has a Legendre transform EðaÞ ¼ inf q2R fqa T ðqÞg. EðaÞ describes the spectrum of local entropies of f . i(ii) If Ka ¼ fx : hlu ðx; f Þ ¼ ag, (lu 6¼ lmax , the measure maximal entropy), then EðaÞ ¼ htop ðf ; Ka Þ. Here htop ðf ; Ka Þ denotes the topological entropy restricted to the set Ka , this is a generalization for non-compact sets of the classical topological entropy [6]. Besides EðaðqÞÞ ¼ qaðqÞ T ðqÞ;
0
aðqÞ :¼ T ðqÞ;
q ¼ E0 ðaÞ:
ð13Þ
Let ai ¼ limq!1 aðqÞ ¼ inf q2R faðqÞg, as ¼ limq!1 aðqÞ ¼ supq2R faðqÞg, then Ka ¼ ;, if a 2 ðai ; as Þ, so that 0 the domain of definition of EðaÞ is the range of T ðqÞ. (iii) From the above variational relationship between EðaÞ and T ðqÞ we have that EðaÞ is concave and continuously differentiable in ðai ; as Þ. For the analysis of the entropies spectrum we consider a dynamical map f : M ! M and a family of potentials U ¼ fuk gk2ðd;dÞ mf ðMÞ. Let define the map sðk; qÞ ¼ P ðquk Þ qP ðuk Þ, where P ¼ P ðuk ; f Þ, k 2 ðd; dÞ. So sð0; qÞ ¼ P ðqu0 Þ qP ðu0 Þ ¼ Tu0 ðqÞ, for the non-perturbed case we shall denote this map directly by T ðqÞ. In the space of functions u : M ! R, we consider the norm kuk ¼ supx2M fjuðxÞjg, and we assume in the next result that the map k ! uk is continuous. Proposition 2. For w, uk 2 mf ðMÞ, k 2 ðd; dÞ, then Z dP ðw þ kuk Þ jk¼0 ¼ u0 dlw ; dk M where lw is an equilibrium state for w.
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Proof. By a property of the topological pressure ([18], Theorem 9.7): jP ðw þ kuk Þ P ðw þ ku0 Þj 6 jkjku0 uk k and by [17] dP ðw þ ku0 Þ jk¼0 ¼ dk
Z u0 dlw :
Therefore: P ðw þ ku0 Þ P ðwÞ ¼ k
Z
u0 dlw þ oðkÞ:
Thus jP ðw þ kuk Þ P ðwÞj ¼ jP ðw þ kuk Þ P ðw þ ku0 Þ þ P ðw þ ku0 Þ P ðwÞj 6 jP ðw þ kuk Þ P ðw þ ku0 Þj þ jP ðw þ ku0 Þ P ðwÞj Z 6 jkjku0 uk k þ jkj u0 dlw þ oðkÞ : Finally P ðw þ kuk Þ P ðwÞ Z 6 ku0 uk k þ oðkÞ; dl u 0 w k
so that
Z dP ðw þ kuk Þ ¼ u0 dlw : dk M k¼0
ð14Þ
To see that the map sðk; qÞ is continuously differentiable with respect to the variable k (the differentiability with respect to q is ensured by the multifractal formalism above described) we may consider it defined from the equation P ðsðk; qÞ quk þ qP ðuk ÞÞ ¼ 0. Now let, P ðk; q; tÞ :¼ P ðt quk þ qP ðuk ÞÞ so we have op jð0; q; sðk; qÞÞ ¼ ot
Z
1 dlq ¼ 1:
ð15Þ
Therefore, by the implicit theorem, sðk; qÞ is well defined in a sufficiently small neighborhood of k ¼ 0, and it is continuously differentiable with respect to k. We assume for the next results in this subsection that the map ðd; dÞ ! UðkÞ, given by k ! uk is of class C 1 . In the next theorem we compute explicitly the derivatives Theorem 3. If UðkÞ ¼ fuk gk2ðd;dÞ is a family of potentials with fuk g mf ðMÞ, then Z osðk; qÞ oUðkÞ oP ðUðkÞÞ ¼ q H dl ; where H :¼ q ok k¼0 ok k¼0 ok k¼0 and lq is an equilibrium state for qu. Proof. We have sðk; qÞ quk þ qP ðuk Þ ¼ T ðqÞ qu0 þ qP ðu0 Þ þ
osðk; qÞ oUðkÞ oP ðUðkÞÞ q k þ oðkÞ: þ q ok k¼0 ok k¼0 ok k¼0
Thus, by Proposition 2, 0¼ So that
Z oP ðsðk; qÞ quk þ qP ðuk ÞÞ osðk; qÞ oUðkÞ oP ðUðkÞÞ q dlq : ¼ ok ok k¼0 ok k¼0 ok k¼0 k¼0
Z osðk; qÞ ¼ q H dlq ; ok k¼0
oUðkÞ oP ðUðkÞÞ H :¼ : ok k¼0 ok k¼0
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We set Tk ðqÞ ¼ sðk; qÞ and we consider the following perturbed spectra: Ek ðaÞ ¼ Tk ðaÞ ¼ inf fqa Tk ðqÞg;
ð16Þ
q2R
Ek ðaðqÞÞ ¼ Tk ðqÞ q
oTk : oq
ð17Þ
If l ¼ lmax , the measure of maximal entropy, then [17] htop if a ¼ htop ; EðaÞ ¼ htop ðf ; Ka Þ ¼ 0 if a 6¼ htop : Taking into account this degenerate behavior when the maximal entropy measure is considered, to ensure the differentiability of the map k ! Ek ðaÞ it should be proved the following: Lemma 4. Let UðkÞ ¼ fuk g mf ðMÞ, if l0 :¼ lu0 6¼ lmax , then lk :¼ luk 6¼ lmax for sufficiently small jkj. S ðuh
Þ
Proof. We use the following result, which is derived from Theorem 3.8 in [17]: lu ¼ lmax if and only if n n top ðxÞ ¼ C for some constant C and for any x 2 Pn ðf Þ ¼ fx : f n ðxÞ ¼ xg, n 2 N. This if l0 :¼ lu0 6¼ lmax , then there exists a n0 2 N and x0 2 Pn0 ðf Þ with Sn0 ðu0 htop Þ ðx0 Þ 6¼ 0: n0
ð18Þ
Let uk 2 mf ðMÞ, such that ku0 uk k <
1 Sn0 ðu0 htop Þ ðx0 Þ: 2 n0
Therefore we have Sn0 ðu0 uk Þ 1 Sn0 ðu0 htop Þ ðx0 Þ 6 ku0 uk k < ðx0 Þ: n0 2 n0
ð19Þ
So that Sn0 ðuk htop Þ Sn ðu htop Þ ðx0 Þ 6¼ 0 for jkj < d :¼ 0 0 ðx0 Þ n0 n0 and hence lk :¼ luk 6¼ lmax for jkj < d.
h
Theorem 5. Let UðkÞ ¼ fuk g mf ðMÞ, if l0 6¼ lmax , then k ! Ek ðaÞ is of class C 1 in a enough small neighborhood of k ¼ 0. Proof. By the results mentioned at the beginning of this section the function q ! sðk; qÞ (for k fixed) is convex and continuously differentiable, so o2 sðk; qÞ=oq2 6¼ 0 and q ! osðk; qÞ=oq is strictly increasing. Then there is a C 1 function uk : R ! R such that osðk;qÞ ðk; uk ðpÞÞ ¼ p. Now, by the multifractal formalism, oq osðk; qÞ osðk; qÞ ðk; qÞ ¼ sðk; qÞ q ; Ek oq oq whereby the above and Theorem 3 the right-side member is of class C 1 . Therefore, by Eq. (16) we have Ek ðpÞ ¼ sðk; uk ðpÞÞ þ puk ðpÞ, so that the spectrum Ek varies in continuously differentiable way in k. h 3.2. Lyapunov spectrum We consider now a perturbation of the dynamical map and we obtain general results in this situation. The specific analysis of the Lyapunov spectrum will arise as a particular case. Let F ¼ ffk gk2ðd;dÞ a family of C 2 diffeomorphisms of a compact manifold M, where f0 is a C 2 diffeomorphism which has the properties of expansiveness and specification. We take firstly a non-perturbed potential u 2 mf0 ðMÞ. We shall study the variational properties of the function sðk; qÞ defined from Pk ðsðk; qÞwðkÞ qu þ qP ðuÞÞ ¼ R0, where Pk ¼ P ðfk ; uÞ and fwðkÞg is a family of functions such that k ! wðkÞ is C 1 . We additionally assume that wð0Þ dlq 6¼ 0, with lq an equilibrium state for qu.
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Like in the case of potential perturbations, before computing the derivatives sðk; qÞ, with respect to k, we justify that really this function is continuously differentiable. This follows similarly to the entropy case, by using the implicit function theorem: Let pðk; q; tÞ :¼ Pk ðtwðkÞ qu þ qP ðuÞÞ, so by Proposition 2, Z op ð0; q; sð0; qÞÞ ¼ wð0Þ dlq 6¼ 0: ot Now the map s is well defined and it has the wished regularity properties. Theorem 6. Let F ¼ ffk gk2ðd;dÞ a family of C 2 diffeomorphisms of a compact manifold M, where f0 has the properties of expansiveness and specification, and k ! fk is a C 2 map. If u 2 mf0 ðMÞ and sðk; qÞ is the function defined as above, then sðk; qÞ is continuously differentiable with respect to k and its derivative is R sð0; qÞ owðkÞ j dlq osðk; qÞ ok k¼0 R ¼ ; ð20Þ ok k¼0 wð0Þ dlq where lq is an equilibrium state for qu. Proof
osðk; qÞ owðkÞ k þ oðkÞ: wð0Þ þ sð0; qÞ sðk; qÞwðkÞ qu þ qP ðuÞ ¼ sð0; qÞ log kDx ðf0 Þk qu þ qP ðuÞ þ ok k¼0 ok k¼0
Then, by applying Proposition 2 0¼
Z Z oP ðsðk; qÞwðkÞ qu þ qP ðuÞÞ osðk; qÞ owðkÞ wð0Þ dl ¼ þ sð0; qÞ dl ; q ok ok k¼0 ok k¼0 q k¼0
so that R sð0; qÞ owðkÞ j dlq osðk; qÞ ok k¼0 R ¼ : ok k¼0 wð0Þ dlq
We may more generally consider families F ¼ ffk gk2ðd;dÞ and U ¼ fuk gk2ðd;dÞ of perturbations of the dynamics and the potential. If sðk; qÞ is defined from Pk ðsðk; qÞwðkÞ quk þ qP ðuk ÞÞ ¼ 0, where Pk ¼ P ðfk ; uk Þ, a simple modification in the demonstration of the above theorem allows to obtain the following variational formula in the more general case when both two, dynamics and potential, are perturbed:
R oUðkÞ R owðkÞ oP ðUðkÞÞ sð0; qÞ q dl dlq j j j k¼0 k¼0 k¼0 q ok ok ok osðk; qÞ R ¼ : ð21Þ ok wð0Þ dlq k¼0 For the specific study of the positive Lyapunov spectrum is considered the following: wðkÞ ¼ wðk; xÞ ¼ log kDx ðfk Þk; if Pk ðWðq; k; UÞÞ ¼ 0 with Wðq; k; UÞ ¼ sðk; qÞwðkÞ quk þ qP ðuk Þ, then for this special case is taken sðk; qÞ defined from Pk ðWðq; k; 0ÞÞ ¼ 0, i.e. Pk ðsðk; qÞwðkÞÞ ¼ qhk , where hk ¼ htop ðfk Þ. To see that ðop=otÞð0; q; sð0; qÞÞ 6¼ 0, notice that by the Birkhoff ergodic theorem (let w0 ¼ wð0Þ) Z Z op 1 1 ð0; q; sð0; qÞÞ ¼ lim w0 ðf0i ðxÞÞ dlq ðxÞ ¼ lim kDx ðf0n Þk: n!1 n n!1 n ot Since the integrand is a member of the Lyapunov spectrum and because the ergodicity of lq , we have that ðop=otÞð0; q; sð0; qÞÞ 6¼ 0, when the strictly positive spectrum is considered. If LD ðaÞ ¼ dimH fx : vðxÞ ¼ ag (dimH is the Hausdorff dimension), then this Lyapunov spectrum can be described by a dimension spectrum, specifically LD ðaðqÞÞ :¼ T 0 ðqÞ ¼ Dlq ðhtop =aðqÞÞ, where Dl ðaÞ ¼ dimH fx : DlðxÞ ¼ ag and Dl is the dimension of the measure (see [14] for details). This dimension spectrum is in turn described by the nonperturbed function T ðqÞ ¼ sð0; qÞ [14], so in the perturbed case we have
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A.M. Meson, F. Vericat / Chaos, Solitons and Fractals 19 (2004) 1031–1038
LD;k ðaðqÞÞ ¼ Dlq ;k
osðk; qÞ ðk; qÞ oq
¼ sðk; qÞ q
osðk; qÞ : oq
ð22Þ
Thus as a particular case of the above description we have that k ! LD;k ðaðqÞÞ is C 1 is a neighborhood of k ¼ 0. We finish with a result related with dimension: let DðkÞ defined from Pk ðDðkÞwðkÞÞ ¼ 0. For conformal expanding dynamical systems with a basic set K, if P ðDð0Þwð0ÞÞ ¼ 0 then Dð0Þ ¼ dimH K [3]. This is a main relationship between the thermodynamic formalism and dimension theory. These systems are special cases of the more general considered along this article. Theorem 7 R Dð0Þ owðkÞ j dlq oDðkÞ ok k¼0 R ¼ : ok k¼0 wð0Þ dlq Proof DðkÞwðkÞ ¼ Dð0Þwð0Þ þ
oDðkÞ owðkÞ k þ oðkÞ: wð0Þ þ Dð0Þ ok k¼0 ok k¼0
By Proposition 2 0¼
Z Z oP ðDðkÞwðkÞÞ oDðkÞ owðkÞ wð0Þ dl ¼ þ Dð0Þ dl q ok ok k¼0 ok k¼0 q k¼0
and so R Dð0Þ owðkÞ j dlq oDðkÞ ok k¼0 R ¼ : ok k¼0 wð0Þ dlq
ð23Þ
Acknowledgements Support of this work by the Consejo Nacional de Investigaciones Cientıficas y Tecnicas (CONICET), Universidad Nacional de La Plata and Agencia Nacional de Promoci on Cintıfica y Tecnol ogica of Argentina (Grants PID 04960,11/ I055 and PICT 03-4517) is greatly appreciated. F.V. is a member of CONICET.
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Benzi R, Paladin G, Parisi G, Vulpiani A. J Phys A 1984;17:3521. Badii R, Polliti A. Phys Scr 1987;35:243. Barreira L, Pesin Y, Schmeling J. Chaos 1997;7(1):27. Barreira L. Nonlinearity 2001;14:259. Bowen R. Trans Am Math Soc 1971;154:377. Bowen R. Trans Am Math Soc 1973;184:125. Brin M, Katok A. In: Geometric dynamics. Lecture notes in mathematics 1007. Springer; 1983. p. 30–8. Colet P, Lebowitz J, Porzio A. J Stat Phys 1987;47:609. Halsey TC, Jensen M, Kadanoff L, Procaccia I, Shraiman B. Phys Rev A 1986;33:1141. Hentschel HGE, Procaccia I. Physica D 1983;8:435. Jensen M. Thesis, Niels Bohr Institute and Nordita, 1994. Katok A, Hasselblatt B. Introduction to the modern theory of dynamical systems. Cambridge University Press; 1995. Martınez V, Paredes S, Borgani S, Coles P. Science 1995;269:1245. Mes on AM, Vericat F. Chaos, Solitons & Fractals 2002;13:1037. Pesin Y, Weiss H. J Stat Phys 1997;86(1–2):233. Ruelle D. Thermodynamic formalism, encyclopedia of mathematics. Addison-Wesley; 1978. Takens F, Verbitski E. Commun Math Phys 1999;203:593. Walters P. Introduction to Ergodic theory. Springer-Verlag; 1982. Weiss H. J Stat Phys 1992;69:879.
Variational analysis for the multifractal spectra of local entropies and Lyapunov exponents n *, Fernando Vericat Alejandro M. Meso Instituto de Fısica de Lıquidos y Sistemas Biol ogicos (IFLYSIB), UNLP-CONICET, cc. 565, 1990 La Plata, Argentina Grupo de Aplicaciones Matem aticas y Estadısticas de la Facultad de Ingenierıa (GAMEFI), Departamento de Fisicomatem atica, Facultad de Ingenierıa, Universidad Nacional de La Plata, Argentina Accepted 22 April 2003 Communicated by I. Procaccia
Abstract In a previous article [Chaos, Solitons and Fractals, 13 (2002) 1037], the authors have analyzed the multifractal Lyapunov spectrum. Here we continue that study by considering perturbations of the potential and the dynamics to obtain variational expressions for the entropies and Lyapunov spectra. The spirit and the framework of this note is to obtain, beyond hyperbolicity, variational results, some of which are new and some other which have already been derived but under stronger conditions. 2003 Elsevier Ltd. All rights reserved.
1. Introduction The concept of ‘‘multifractal analysis’’ was raised in the physical ambient [1,2]. One of the main motivations was the study of the behavior of physical measures on strange attractors, as well as aspects related with diffusion-limited, aggregates, etc. [9]. Among another areas of interest it could be mentioned: the study of turbulence [11], distribution and cluster of galaxies [13] and some subjects in Biology. In the study of chaotic behaviors are frequently found invariant sets with a complex mathematical structure. The multifractal analysis essentially treats with decompositions, the so-called fractal decomposition, of these sort of sets. Despite the complexity of the level sets, in many cases it can be found a well comported function which encodes the information about the decomposition. Let consider for instance a decomposition of an attractor A, carrying an invariant measure l, in sets Ka ¼ f x : lðBr ðxÞÞ ra as r ! 0g; ð1Þ where Br ðxÞ is the ball of centre x and radius r. The function fa ðxÞ ¼ dimH Ka , the Haussdorf dimension of the set Ka , was studied in [8,10], these approaches were generalized to study local (depending on points) characteristics of dynamical systems. A complete description of the multifractal analysis of invariant measures was performed by Pesin and Weiss in [15]. In this work were extended all the results known until that moment about smooth conformal maps. The general concept of multifractal analysis was introduced in [3], we present here a brief account: Let X be a set and g : X ! ½1; þ1, now the sets in which X is partitioned are Ka ¼ Ka ðgÞ ¼ fx : gðxÞ ¼ ag:
ð2Þ
* Corresponding author. Address: Instituto de Fısica de Lıquidos y Sistemas Biol ogicos (IFLYSIB), UNLP-CONICET, cc. 565, 1990 La Plata, Argentina. Fax: +54-221-425-7317. E-mail addresses: meson@iflysib.unlp.edu.ar (A.M. Mes on), vericat@iflysib.unlp.edu.ar (F. Vericat).
0960-0779/$ - see front matter 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0960-0779(03)00204-2
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Let G be a function defined on sets, and let F ðaÞ ¼ GðKa Þ, the map F is called the multifractal spectrum specified by the pair ðg; GÞ. Examples of such multifractal spectra are gðxÞ ¼ Dl ðxÞ, the pointwise dimension of the measure l, gðxÞ ¼ hðf ; xÞ, the local entropy or gðxÞ ¼ kðxÞ, Lyapunov exponents. In the paper by Barreira et al. [3], a complete multifractal analysis for pointwise dimensions, local entropies as well as for Lyapunov exponents of conformal expanding maps and Axiom A-diffeomorphisms was performed. Accordingly, for these dynamical systems, the existence of Markov partitions and Gibbs measures is ensured. Results of that article, concerning to local pointwise entropies, were recovered, under weaker hypothesis, by Takens and Verbitski in [17]. In [14] we gave a description of the Lyapunov spectrum, under hyphothesis like Takens–Verbitski. In this article we study the variational properties of perturbations on the entropies and Lyapunov spectra. More specifically we shall consider firstly, for a fixed dynamical map f : X ! X a family of perturbative potentials with respect to u0 : U ¼ fuk gk2ðd;dÞ and study the variation of the entropies spectrum, by analyzing the derivatives of function sðk; qÞ which is in turn is a perturbation of the function T ðqÞ ¼ P ðqu0 Þ qP ðu0 Þ (P is the topological pressure) whose Legendre transformation describes the spectrum of local entropies for the non-perturbative case [17]. For the Lyapunov spectrum we consider more general perturbations, besides a family of potentials we introduce a family of dynamics F ¼ ffk gk2ðd;dÞ . According to [14], the perturbative functions to be considered must be sðk; qÞ, which is determined from P ðqhtop ðfk Þ sðk; qÞwðkÞÞ ¼ 0;
ð3Þ
with wðk; xÞ :¼ log kDx ðfk Þk. Here the topological pressure is that associated to uk and fk . The study of the effect of these perturbations is very important for numerical computations, the numerical results could be affected by small perturbations and it may be interesting to know the influence of the perturbations. Variational results about dimension spectra were obtained in [4,19] for hyperbolic diffeomorphisms. In this article we shall derive variational formula, but under much weaker hyphothesis. The paper is organized as follows. In next section we recall some definitions and relevant results which will be useful for further demonstrations. There we also declare our hypothesis. In Section 3 we announce and prove our results.
2. Previous definitions Let M be a compact metric space, and f : M ! M a continuous map. A function u : M ! R, will be called a ‘‘potential’’. The topological pressure associated to a given potential u is the number [12–18]: Z P ðuÞ ¼ sup hl ðf Þ þ u dl ; ð4Þ l
where the supreme is taken over all the Borel measures l, f ––invariant on M, and where hl ðf Þ is the usual Kolmogorov measure-theoretic entropy of f . An equilibrium state for the potential u is a measure lu which satisfies: Z ð5Þ P ðuÞ ¼ hlu ðf Þ þ u dlu : Under certain conditions imposed on the map f and the potential u an equilibrium state can be constructed [12–17]. The specification property for a map f : M ! M intuitively says that, for specified trajectory segments, a periodic orbit that approximates them can be found. This condition ensures abundance of periodic points. The formal definition of this concept, as it was introduced by Bowen in [5], reads: A homeomorphism f : M ! M have the specification property if given a finite disjoint collection of integer intervals I1 ; . . . ; Ik then, for e > 0, there is a integer MðeÞ and a function U : I ¼ [Ii ! M such that (i) distðIi ; Ij Þ > MðeÞ (Euclidean distance) (ii) f n1 n2 ðUðn1 ÞÞ ¼ Uðn2 Þ (iii) dðf n ðxÞ; UðnÞÞ < e, for some x : f m ðxÞ ¼ x, with m P MðeÞ þ lengthðIÞ and for every n 2 I. A homeomorphism f : M ! M is called expansive if there is a constant d > 0, such that dðf n ðxÞ; f n ðyÞÞ < d, for any integer n implies x ¼ y. Notice that both definitions (about specification and expansiveness) were done without using a differentiable structure. They are topological concepts.
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The following metric in M, is derived from the original one d: dn ðx; yÞ ¼ max fdðf i ðxÞ; f i ðyÞÞ : i ¼ 0; 1; . . . ; n 1g:
ð6Þ
The ball of centre x and radius e in the metric dn will be denoted by Bn;e ðxÞ. For a potential u we put Sn ðuÞðxÞ ¼
n1 X
uðf i ðxÞÞ:
ð7Þ
i¼0
Following [12] or [17], we say that a potential u belongs to the class mf ðMÞ if this condition is fulfilled: There are constants e, K > 0 in a such way that dn ðx; yÞ < e ) jSn ðuÞðxÞ Sn ðuÞðyÞj < K:
ð8Þ
Now we recall how a measure can be defined in such a way that it results to be an equilibrium state: Let Pn ðf Þ ¼ fx : f n ðxÞ ¼ xg, set for a potential u: lu;n ¼
X 1 expðSn ðuÞðxÞÞdx ; N ðf ; u; nÞ x2P ðf Þ
ð9Þ
n
P where N ðf ; u; nÞ ¼ x2Pn ðf Þ expðSn ðuÞðxÞÞ and dx is the unit measure at x. The weak limit of the sequence flu;n g, is an equilibrium state lu , i.e. Z Z gðxÞ dlu;n ¼ gðxÞ dlu ; lim n!1
ð10Þ
for every continuous gðxÞ [12–16]. A measure l on M is called a Gibbs state if, for sufficiently small e > 0, there are constants Ae , Be > 0, such that, for any x 2 M and for any positive integer n, is Ae ðexpðSn ðuÞðxÞÞ nP ðuÞÞ 6 lðBn;e ðxÞÞ 6 Be ðexpðSn ðuÞðxÞÞ nP ðuÞÞ:
ð11Þ
Theorem 1 [12–16]. Let f be an expansive homeomorphism which have the specification property and u a potential belonging to the class mf ðMÞ, then lu is an equilibrium state associated to u, which is a Gibbs state. Besides it is ergodic. We recall the definition of Lyapunov exponents. Let M be a compact Riemannian differentiable manifold d–– dimensional. Let f : M ! M be a C 2 diffeomorphism; we denote by Lx the differential map of f in x, and by Lnx the differential map of f n in x. If v is a tangent vector to M in x, let vðxÞ ¼ vðx; v; f Þ :¼ lim
n!1
1 log Lnx v: n
ð12Þ
The number vðxÞ ¼ vðx; v; f Þ is called the Lyapunov exponent with respect to L ¼ Lðf Þ in ðx; vÞ, provided the limit exists. For a real number v and x 2 M, we denote Ev ðxÞ ¼ fv : vðx; v; f Þ 6 vg: Notice that if v1 6 v2 then Ev1 ðxÞ Ev2 ðxÞ. Furthermore, if x 2 M then there exists an integer mðxÞ 6 d, such that there is a collection of numbers v1 ðxÞ; v2 ðxÞ; . . . ; vmðxÞ ðxÞ and linear subspaces Ev1 ðxÞ; Ev2 ðxÞ; . . . ; EvmðxÞ ðxÞ, with k1 ðxÞ < k2 ðxÞ < < kmðxÞ ðxÞ; Ek1 ðxÞ Ek2 ðxÞ EkmðxÞ ðxÞ: If v 2 Eviþ1 ðxÞ Evi ðxÞ then vðx; vÞ ¼ viþ1 ðxÞ. The Lyapunov spectrum is now defined as Spx ðL; f Þ ¼ fvi ðxÞ : i ¼ 1; 2; . . . ; mðxÞg:
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Our main hypothesis will be (a) For the entropies spectrum f : M ! M, with M a compact metric space, an expansive homeomorphism with specification property. (b) For the Lyapunov spectrum • f : M ! M a C 2 expansive diffeomorphism and with specification property, M a compact Riemannian manifold. • The differential map of f satisfies: there are constants e, K > 0 in a such way that dn ðx; yÞ < e ) Lnx Lny < K; so that the potential wðxÞ ¼ log kLx k belongs to the class mf ðMÞ. Indeed the statistical sum Sn ðwÞðxÞ is equal to n1 X
n1 Y Li ¼ log Ln : log Lix ¼ log x x
i¼0
i¼0
To finish this section we recall the following definition: let f : M ! M be a continuous map and M a compact metric space. The upper and lower local entropies are hl ðx; f Þ ¼ lim lim sup e!0 n!1
hl ðx; f Þ ¼ lim lim inf e!0 n!1
1 log lðBe;n ðxÞÞ; n
1 log lðBe;n ðxÞÞ: n
Then (Brin–Katok theorem [7]), the local entropy does exist, i.e. hl ðx; f Þ ¼ hl ðx; f Þ :¼ hl ðx; f Þ, for every x 2 X .
3. Results 3.1. Entropies spectrum We begin by recalling the main facts of the multifractal formalism developed by Takens and Verbitski [17]. ii(i) The function T ðqÞ ¼ Tu ðqÞ :¼ P ðquÞ qP ðuÞ, u 2 mf ðMÞ, is convex and continuously differentiable with T 0 ðqÞ ¼ R u dlq , where lq is an equilibrium state for qu. This map has a Legendre transform EðaÞ ¼ inf q2R fqa T ðqÞg. EðaÞ describes the spectrum of local entropies of f . i(ii) If Ka ¼ fx : hlu ðx; f Þ ¼ ag, (lu 6¼ lmax , the measure maximal entropy), then EðaÞ ¼ htop ðf ; Ka Þ. Here htop ðf ; Ka Þ denotes the topological entropy restricted to the set Ka , this is a generalization for non-compact sets of the classical topological entropy [6]. Besides EðaðqÞÞ ¼ qaðqÞ T ðqÞ;
0
aðqÞ :¼ T ðqÞ;
q ¼ E0 ðaÞ:
ð13Þ
Let ai ¼ limq!1 aðqÞ ¼ inf q2R faðqÞg, as ¼ limq!1 aðqÞ ¼ supq2R faðqÞg, then Ka ¼ ;, if a 2 ðai ; as Þ, so that 0 the domain of definition of EðaÞ is the range of T ðqÞ. (iii) From the above variational relationship between EðaÞ and T ðqÞ we have that EðaÞ is concave and continuously differentiable in ðai ; as Þ. For the analysis of the entropies spectrum we consider a dynamical map f : M ! M and a family of potentials U ¼ fuk gk2ðd;dÞ mf ðMÞ. Let define the map sðk; qÞ ¼ P ðquk Þ qP ðuk Þ, where P ¼ P ðuk ; f Þ, k 2 ðd; dÞ. So sð0; qÞ ¼ P ðqu0 Þ qP ðu0 Þ ¼ Tu0 ðqÞ, for the non-perturbed case we shall denote this map directly by T ðqÞ. In the space of functions u : M ! R, we consider the norm kuk ¼ supx2M fjuðxÞjg, and we assume in the next result that the map k ! uk is continuous. Proposition 2. For w, uk 2 mf ðMÞ, k 2 ðd; dÞ, then Z dP ðw þ kuk Þ jk¼0 ¼ u0 dlw ; dk M where lw is an equilibrium state for w.
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Proof. By a property of the topological pressure ([18], Theorem 9.7): jP ðw þ kuk Þ P ðw þ ku0 Þj 6 jkjku0 uk k and by [17] dP ðw þ ku0 Þ jk¼0 ¼ dk
Z u0 dlw :
Therefore: P ðw þ ku0 Þ P ðwÞ ¼ k
Z
u0 dlw þ oðkÞ:
Thus jP ðw þ kuk Þ P ðwÞj ¼ jP ðw þ kuk Þ P ðw þ ku0 Þ þ P ðw þ ku0 Þ P ðwÞj 6 jP ðw þ kuk Þ P ðw þ ku0 Þj þ jP ðw þ ku0 Þ P ðwÞj Z 6 jkjku0 uk k þ jkj u0 dlw þ oðkÞ : Finally P ðw þ kuk Þ P ðwÞ Z 6 ku0 uk k þ oðkÞ; dl u 0 w k
so that
Z dP ðw þ kuk Þ ¼ u0 dlw : dk M k¼0
ð14Þ
To see that the map sðk; qÞ is continuously differentiable with respect to the variable k (the differentiability with respect to q is ensured by the multifractal formalism above described) we may consider it defined from the equation P ðsðk; qÞ quk þ qP ðuk ÞÞ ¼ 0. Now let, P ðk; q; tÞ :¼ P ðt quk þ qP ðuk ÞÞ so we have op jð0; q; sðk; qÞÞ ¼ ot
Z
1 dlq ¼ 1:
ð15Þ
Therefore, by the implicit theorem, sðk; qÞ is well defined in a sufficiently small neighborhood of k ¼ 0, and it is continuously differentiable with respect to k. We assume for the next results in this subsection that the map ðd; dÞ ! UðkÞ, given by k ! uk is of class C 1 . In the next theorem we compute explicitly the derivatives Theorem 3. If UðkÞ ¼ fuk gk2ðd;dÞ is a family of potentials with fuk g mf ðMÞ, then Z osðk; qÞ oUðkÞ oP ðUðkÞÞ ¼ q H dl ; where H :¼ q ok k¼0 ok k¼0 ok k¼0 and lq is an equilibrium state for qu. Proof. We have sðk; qÞ quk þ qP ðuk Þ ¼ T ðqÞ qu0 þ qP ðu0 Þ þ
osðk; qÞ oUðkÞ oP ðUðkÞÞ q k þ oðkÞ: þ q ok k¼0 ok k¼0 ok k¼0
Thus, by Proposition 2, 0¼ So that
Z oP ðsðk; qÞ quk þ qP ðuk ÞÞ osðk; qÞ oUðkÞ oP ðUðkÞÞ q dlq : ¼ ok ok k¼0 ok k¼0 ok k¼0 k¼0
Z osðk; qÞ ¼ q H dlq ; ok k¼0
oUðkÞ oP ðUðkÞÞ H :¼ : ok k¼0 ok k¼0
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We set Tk ðqÞ ¼ sðk; qÞ and we consider the following perturbed spectra: Ek ðaÞ ¼ Tk ðaÞ ¼ inf fqa Tk ðqÞg;
ð16Þ
q2R
Ek ðaðqÞÞ ¼ Tk ðqÞ q
oTk : oq
ð17Þ
If l ¼ lmax , the measure of maximal entropy, then [17] htop if a ¼ htop ; EðaÞ ¼ htop ðf ; Ka Þ ¼ 0 if a 6¼ htop : Taking into account this degenerate behavior when the maximal entropy measure is considered, to ensure the differentiability of the map k ! Ek ðaÞ it should be proved the following: Lemma 4. Let UðkÞ ¼ fuk g mf ðMÞ, if l0 :¼ lu0 6¼ lmax , then lk :¼ luk 6¼ lmax for sufficiently small jkj. S ðuh
Þ
Proof. We use the following result, which is derived from Theorem 3.8 in [17]: lu ¼ lmax if and only if n n top ðxÞ ¼ C for some constant C and for any x 2 Pn ðf Þ ¼ fx : f n ðxÞ ¼ xg, n 2 N. This if l0 :¼ lu0 6¼ lmax , then there exists a n0 2 N and x0 2 Pn0 ðf Þ with Sn0 ðu0 htop Þ ðx0 Þ 6¼ 0: n0
ð18Þ
Let uk 2 mf ðMÞ, such that ku0 uk k <
1 Sn0 ðu0 htop Þ ðx0 Þ: 2 n0
Therefore we have Sn0 ðu0 uk Þ 1 Sn0 ðu0 htop Þ ðx0 Þ 6 ku0 uk k < ðx0 Þ: n0 2 n0
ð19Þ
So that Sn0 ðuk htop Þ Sn ðu htop Þ ðx0 Þ 6¼ 0 for jkj < d :¼ 0 0 ðx0 Þ n0 n0 and hence lk :¼ luk 6¼ lmax for jkj < d.
h
Theorem 5. Let UðkÞ ¼ fuk g mf ðMÞ, if l0 6¼ lmax , then k ! Ek ðaÞ is of class C 1 in a enough small neighborhood of k ¼ 0. Proof. By the results mentioned at the beginning of this section the function q ! sðk; qÞ (for k fixed) is convex and continuously differentiable, so o2 sðk; qÞ=oq2 6¼ 0 and q ! osðk; qÞ=oq is strictly increasing. Then there is a C 1 function uk : R ! R such that osðk;qÞ ðk; uk ðpÞÞ ¼ p. Now, by the multifractal formalism, oq osðk; qÞ osðk; qÞ ðk; qÞ ¼ sðk; qÞ q ; Ek oq oq whereby the above and Theorem 3 the right-side member is of class C 1 . Therefore, by Eq. (16) we have Ek ðpÞ ¼ sðk; uk ðpÞÞ þ puk ðpÞ, so that the spectrum Ek varies in continuously differentiable way in k. h 3.2. Lyapunov spectrum We consider now a perturbation of the dynamical map and we obtain general results in this situation. The specific analysis of the Lyapunov spectrum will arise as a particular case. Let F ¼ ffk gk2ðd;dÞ a family of C 2 diffeomorphisms of a compact manifold M, where f0 is a C 2 diffeomorphism which has the properties of expansiveness and specification. We take firstly a non-perturbed potential u 2 mf0 ðMÞ. We shall study the variational properties of the function sðk; qÞ defined from Pk ðsðk; qÞwðkÞ qu þ qP ðuÞÞ ¼ R0, where Pk ¼ P ðfk ; uÞ and fwðkÞg is a family of functions such that k ! wðkÞ is C 1 . We additionally assume that wð0Þ dlq 6¼ 0, with lq an equilibrium state for qu.
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1037
Like in the case of potential perturbations, before computing the derivatives sðk; qÞ, with respect to k, we justify that really this function is continuously differentiable. This follows similarly to the entropy case, by using the implicit function theorem: Let pðk; q; tÞ :¼ Pk ðtwðkÞ qu þ qP ðuÞÞ, so by Proposition 2, Z op ð0; q; sð0; qÞÞ ¼ wð0Þ dlq 6¼ 0: ot Now the map s is well defined and it has the wished regularity properties. Theorem 6. Let F ¼ ffk gk2ðd;dÞ a family of C 2 diffeomorphisms of a compact manifold M, where f0 has the properties of expansiveness and specification, and k ! fk is a C 2 map. If u 2 mf0 ðMÞ and sðk; qÞ is the function defined as above, then sðk; qÞ is continuously differentiable with respect to k and its derivative is R sð0; qÞ owðkÞ j dlq osðk; qÞ ok k¼0 R ¼ ; ð20Þ ok k¼0 wð0Þ dlq where lq is an equilibrium state for qu. Proof
osðk; qÞ owðkÞ k þ oðkÞ: wð0Þ þ sð0; qÞ sðk; qÞwðkÞ qu þ qP ðuÞ ¼ sð0; qÞ log kDx ðf0 Þk qu þ qP ðuÞ þ ok k¼0 ok k¼0
Then, by applying Proposition 2 0¼
Z Z oP ðsðk; qÞwðkÞ qu þ qP ðuÞÞ osðk; qÞ owðkÞ wð0Þ dl ¼ þ sð0; qÞ dl ; q ok ok k¼0 ok k¼0 q k¼0
so that R sð0; qÞ owðkÞ j dlq osðk; qÞ ok k¼0 R ¼ : ok k¼0 wð0Þ dlq
We may more generally consider families F ¼ ffk gk2ðd;dÞ and U ¼ fuk gk2ðd;dÞ of perturbations of the dynamics and the potential. If sðk; qÞ is defined from Pk ðsðk; qÞwðkÞ quk þ qP ðuk ÞÞ ¼ 0, where Pk ¼ P ðfk ; uk Þ, a simple modification in the demonstration of the above theorem allows to obtain the following variational formula in the more general case when both two, dynamics and potential, are perturbed:
R oUðkÞ R owðkÞ oP ðUðkÞÞ sð0; qÞ q dl dlq j j j k¼0 k¼0 k¼0 q ok ok ok osðk; qÞ R ¼ : ð21Þ ok wð0Þ dlq k¼0 For the specific study of the positive Lyapunov spectrum is considered the following: wðkÞ ¼ wðk; xÞ ¼ log kDx ðfk Þk; if Pk ðWðq; k; UÞÞ ¼ 0 with Wðq; k; UÞ ¼ sðk; qÞwðkÞ quk þ qP ðuk Þ, then for this special case is taken sðk; qÞ defined from Pk ðWðq; k; 0ÞÞ ¼ 0, i.e. Pk ðsðk; qÞwðkÞÞ ¼ qhk , where hk ¼ htop ðfk Þ. To see that ðop=otÞð0; q; sð0; qÞÞ 6¼ 0, notice that by the Birkhoff ergodic theorem (let w0 ¼ wð0Þ) Z Z op 1 1 ð0; q; sð0; qÞÞ ¼ lim w0 ðf0i ðxÞÞ dlq ðxÞ ¼ lim kDx ðf0n Þk: n!1 n n!1 n ot Since the integrand is a member of the Lyapunov spectrum and because the ergodicity of lq , we have that ðop=otÞð0; q; sð0; qÞÞ 6¼ 0, when the strictly positive spectrum is considered. If LD ðaÞ ¼ dimH fx : vðxÞ ¼ ag (dimH is the Hausdorff dimension), then this Lyapunov spectrum can be described by a dimension spectrum, specifically LD ðaðqÞÞ :¼ T 0 ðqÞ ¼ Dlq ðhtop =aðqÞÞ, where Dl ðaÞ ¼ dimH fx : DlðxÞ ¼ ag and Dl is the dimension of the measure (see [14] for details). This dimension spectrum is in turn described by the nonperturbed function T ðqÞ ¼ sð0; qÞ [14], so in the perturbed case we have
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LD;k ðaðqÞÞ ¼ Dlq ;k
osðk; qÞ ðk; qÞ oq
¼ sðk; qÞ q
osðk; qÞ : oq
ð22Þ
Thus as a particular case of the above description we have that k ! LD;k ðaðqÞÞ is C 1 is a neighborhood of k ¼ 0. We finish with a result related with dimension: let DðkÞ defined from Pk ðDðkÞwðkÞÞ ¼ 0. For conformal expanding dynamical systems with a basic set K, if P ðDð0Þwð0ÞÞ ¼ 0 then Dð0Þ ¼ dimH K [3]. This is a main relationship between the thermodynamic formalism and dimension theory. These systems are special cases of the more general considered along this article. Theorem 7 R Dð0Þ owðkÞ j dlq oDðkÞ ok k¼0 R ¼ : ok k¼0 wð0Þ dlq Proof DðkÞwðkÞ ¼ Dð0Þwð0Þ þ
oDðkÞ owðkÞ k þ oðkÞ: wð0Þ þ Dð0Þ ok k¼0 ok k¼0
By Proposition 2 0¼
Z Z oP ðDðkÞwðkÞÞ oDðkÞ owðkÞ wð0Þ dl ¼ þ Dð0Þ dl q ok ok k¼0 ok k¼0 q k¼0
and so R Dð0Þ owðkÞ j dlq oDðkÞ ok k¼0 R ¼ : ok k¼0 wð0Þ dlq
ð23Þ
Acknowledgements Support of this work by the Consejo Nacional de Investigaciones Cientıficas y Tecnicas (CONICET), Universidad Nacional de La Plata and Agencia Nacional de Promoci on Cintıfica y Tecnol ogica of Argentina (Grants PID 04960,11/ I055 and PICT 03-4517) is greatly appreciated. F.V. is a member of CONICET.
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