Multifractal Analysis of Local Entropies for Gibbs Measures

Recently in the series of papers L. Barreira, Ya.B. Pesin, J. Schmeling and H. Weiss performed a complete multifractal analysis of local dimensions, entropies and Lyapunov exponents of conformal expanding maps and surface Axion A diffeomorphisms for Gibbs measures. The main goal of these papers was primarily the analysis of the local (pointwise) dimensions. This is an extremely difficult problem and, for example, similar results for hyperbolic systems in dimensions 3 and higher have not been yet obtained. In the present work we concentrate our attention on the multifractal analysis of the local (pointwise) entropies. We are able to obtain results, which are similar to those mentioned above, for Gibbs measures of the expansive homeomorphisms with specification property. Note that such homeomorphisms may not have Markov partitions, which is a crucial condition in the previous works. However, due to the fact that less is known about thermodynamical properties of these dynamical systems we were able to obtain only the continuous differentiability of the multifractal spectrum of local entropies (compare: the same spectra for the dynamical systems with Markov partitions are analytic). We believe that the smoothness of the multifractal spectrum in our case can be improved. We have related the multifractal spectrum of the local entropies to the the spectrum of correlation entropies. These correlation entropies serve as entopy-like analogs of the Hentshel-Procaccia and Rényi spectra of generalized dimensions. This allows us to complete the duality between the multifractal analyses of local dimensions and entropies. Complete proofs can be found in [TV98] and will appear elsewhere.     

Sets of “Non-typical” points have full topological entropy and full Hausdorff dimension

For subshifts of finite type, conformal repellers, and conformal horseshoes, we prove that the set of points where the pointwise dimensions, local entropies, Lyapunov exponents, and Birkhoff averages do not exist simultaneously, carries full topological entropy and full Hausdorff dimension. This follows from a much stronger statement formulated for a class of symbolic dynamical systems which includes subshifts with the specification property. Our proofs strongly rely on the multifractal analysis of dynamical systems and constitute a non-trivial mathematical application of this theory.     

Multifractal analysis of local entropies for recurrence time

We introduce local entropies and multifractal spectra associated with Poincaré recurrences. By using characteristics of a dynamical system we establish an exact formula on multifractal spectrum of local entropies for recurrence time.     

Dimension Theory for Multimodal Maps

This paper is devoted to the study of dimension theory, in particular multifractal analysis, for multimodal maps. We describe the Lyapunov spectrum and study the multifractal spectrum of pointwise dimension. The lack of regularity of the thermodynamic formalism for this class of maps is reflected in the phase transitions of the spectra.     

局部熵高维重分形谱的上界估计

We discuss the problem of higher-dimensional multifractal spectrum of local entropy for arbitrary invariant measures. By utilizing characteristics of a dynamical system, namely, higher-dimensional entropy capacities and higher-dimensional correlation entropies, we obtain three upper estimates on the hlgher-dimensional multifractal spectrum of local entropies. We also study the domain of higher-dimensional multifractai spetrum of entropies.     

Multifractal and Multi-Multifractal of One-Dimensional Expanding Markov Maps

In measure theory, one is interested in local behaviours, for example in local dimensions, local entropies or local Lyapunov exponents. It has been relevant to study dynamical systems where one can develop further the study of multifractal and multi-multifractal, particularly when there exist strange attractors or repellers. Multifractal and multi-multifractal refer to a notion of size, which emphasizes the local variations of different values coming from the theory of dynamical systems and generated by the dimension theory of invariant measures. This paper gives some part of the literature in this field. Many results are already known, but the large deviations approach allows us to reprove these results and to obtain quite easily results concerning extremal points and extremal measures.     

非自治动力系统的原像熵

本文对紧致度量空间上的连续自映射序列应用生成集和分离集引入了点原像熵、原像分枝熵以及原像关系熵等几类原像熵的定义并进行了研究.主要结果是:(1) 证明了这些熵都是等度拓扑共轭不变量.(2)讨论了这些原像熵之间及它们与拓扑熵之间的关系,得到了联系这些熵的不等式.(3)证明了对正向可扩的连续自映射序列而言, 两类点原像熵相等,原像分枝熵与原像关系熵也相等.(4)证明了对(a).由闭Riemann 流形上的一个扩张映射经充分小的C1-扰动生成的自映射序列,以及(b).有限图上等度连续的自映射序列,有零原像分枝熵.     

Bowen entropy for actions of amenable groups

Bowen introduced a definition of topological entropy of subset inspired by Hausdorff dimension in 1973 [1]. In this paper we consider the Bowen entropy for amenable group action dynamical systems and show that, under the tempered condition, the Bowen entropy of the whole compact space for a given Følner sequence equals the topological entropy. For the proof of this result, we establish a variational principle related to the Bowen entropy and the Brin–Katok local entropy formula for dynamical systems with amenable group actions.     

Ergodic properties for some non-expanding non-reversible systems

We study several properties of invariant measures obtained from preimages, for non-invertible maps on fractal sets which model non-reversible dynamical systems. We give two ways to describe the distribution of all preimages for endomorphisms which are not necessarily expanding on a basic set Λ. We give a topological dynamics condition which guarantees that the corresponding measures converge to a unique conformal ergodic borelian measure; this helps in estimating the unstable dimension a.e. with respect to this measure with the help of Lyapunov exponents. When there exist negative Lyapunov exponents of this limit measure, we study the conditional probabilities induced on the non-uniform local stable manifolds by the limit measure, and also its pointwise dimension on stable manifolds.     

A Family of Functions with Two Different Spectra of Singularities

Our goal is to study the multifractal properties of functions of a given family which have few non vanishing wavelet coefficients. We compute at each point the pointwise Hölder exponent of these functions and also their local \(L^p\) regularity, computing the so-called \(p\) -exponent. We prove that in the general case the Hölder and \(p\) -exponent are different at each point. We also compute the dimension of the sets where the functions have a given pointwise regularity and prove that these functions are multifractal both from the point of view of Hölder and \(L^p\) local regularity with different spectra of singularities. Furthermore, we check that multifractal formalism type formulas hold for functions in that family.     

A new chaotic attractor

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.     

Entropy transmission in C1-dynamical systems

Three entropies of a state in C¹-dynamical systems are introduced and their relations and dynamical properties are studied. The entropy (information) transmission under a channel between two dynamical systems is considered. We find a condition under which our entropy becomes a dynamical invariant between two systems.     

Local variational principle concerning entropy of a sofic group action

Recently Lewis Bowen introduced a notion of entropy for measure-preserving actions of countable sofic groups admitting a generating measurable partition with finite entropy; and then David Kerr and Hanfeng Li developed an operator-algebraic approach to actions of countable sofic groups not only on a standard probability space but also on a compact metric space, and established the global variational principle concerning measure-theoretic and topological entropy in this sofic context. By localizing these two kinds of entropy, in this paper we prove a local version of the global variational principle for any finite open cover of the space, and show that these local measure-theoretic and topological entropies coincide with their classical counterparts when the acting group is an infinite amenable group.     

Multifractal Formalism for Self-Similar Functions Under the Action of Nonlinear Dynamical Systems

We study functions which are self-similar under the action of some nonlinear dynamical systems. We compute the exact pointwise H{?}lder regularity, then we determine the spectrum of singularities and the Besov ``smoothness' index, and finally we prove the multifractal formalism. The main tool in our computation is the wavelet analysis. October 1, 1996. Date revised: May 13, 1997. Date re-revised: January 10, 1998. Date accepted: February 27, 1998.     

Measure theoretical entropy of covers

In this paper we introduce three notions of measure theoretical entropy of a measurable cover in a measure theoretical dynamical system. Two of them were already introduced in [R] and the new one is defined only in the ergodic case. We then prove that these three notions coincide, thus answering a question posed in [R], and recover a variational inequality (proved in [GW]) and a proof of the classical variational principle based on a comparison between the entropies of covers and partitions.     

Infinite non-conformal iterated function systems

We consider a class of iterated function systems consisting of a countable infinity of non-conformal contractions, extending both the self-affine limit sets of Lalley and Gatzouras as well as the infinite iterated function systems of Mauldin and Urbański. Natural examples include the sets of points in the plane obtained by taking the binary expansion along the vertical and the continued fraction expansion along the horizontal and deleting certain pairs of digits. We prove that the Hausdorff dimension of the limit set is equal to the supremum of the dimensions of compactly supported ergodic measures, which are given by a Ledrappier and Young type formula. In addition we consider the multifractal analysis of Birkhoff averages for countable families of potentials. We obtain a conditional variational principle for the level sets.     

DLA with two species: Renormalization-group method renormalization-group method

In this paper, we have studied the structure of DLA with two species by using the kinetic real-space renormalization group method introduced by Wang. Following the RG rules and growth processor, We have gained the configuration of 2×2 cell, calculated the fractal dimensions, multifractal spectra, and free energy when different value of p are applied. And we studied the problem of phase transition with different value of p. Our results demonstrate that the change of p doesn't affect the fractal dimension, but can affect the multifractal spectrum and the phase transition.     

Multifractal Components of Multiplicative Set Functions

We analyze the multifractal spectrumof multiplicative set functions on a self-similar set with open set condition. We show that the multifractal components carry self-similar measures which maximize the dimension. This gives the dimension of a multifractal component as the solution of a problem of maximization of a quasiconcave function satisfying a set of linear constraints. Our analysis covers the case of multifractal components of self-similar measures, the case of Besicovitch normal sets of points, the multifractal spectrum of the relative logarithmic density of a pair of self-similar measures, the multifractal spectrum of the Liapunov exponent of the shift mapping and the intersections of all these sets. We show that the dimension of an arbitrary union of multifractal components is the supremum of the dimensions of the multifractal components in the union. The multidimensional Legendre transform is introduced to obtain the dimension of the intersection of finitely many multifractal components.     

Devil’s carpet of topological entropy and complexity of global dynamical behavior

For bimodal maps the concept of an equal topological entropy class (ETEC) is established by the dual star products. All the infinitely many ETEC plateaus and single points are harmonically organized in the kneading parameter plane, they construct a multifractal devil’s carpet, which possesses a perfect subregion similarity and a dual central symmetry. The entropy devil’s carpet reveals the complexity of global dynamical behavior in the whole parameter plane of bimodal systems.     

Local and pointwise error estimates of the local discontinuous Galerkin method applied to the Stokes problem

We prove local and pointwise error estimates for the local discontinuous Galerkin method applied to the Stokes problem in two and three dimensions. By using techniques originally developed by A. Schatz [Math. Comp., 67 (1998), 877-899] to prove pointwise estimates for the Laplace equation, we prove optimal weighted pointwise estimates for both the velocity and the pressure for domains with smooth boundaries.