A multifractal analysis of equilibrium measures for conformal expanding maps and Moran-like geometric constructions

In this paper we establish the complete multifractal formalism for equilibrium measures for Hölder continuous conformal expanding maps andexpanding Markov Moran-like geometric constructions. Examples include Markov maps of an interval, beta transformations of an interval, rational maps with hyperbolic Julia sets, and conformal toral endomorphisms. We also construct a Hölder continuous homeomorphism of a compact metric space with an ergodic invariant measure of positive entropy for which the dimension spectrum is not convex, and hence the multifractal formalism fails.     

On the Hausdorff Dimension of Invariant Measures of Weakly Contracting on Average Measurable IFS

We consider measures which are invariant under a measurable iterated function system with positive, place-dependent probabilities in a separable metric space. We provide an upper bound of the Hausdorff dimension of such a measure if it is ergodic. We also prove that it is ergodic iff the related skew product is.     

Hausdorff dimensions in two-dimensional maps and thermodynamic formalism

We compute numerically the Hausdorff dimensions of the Gibbs measures on the invariant sets of Axiom A systems. In particular, we stress the existence of a measure which has maximal dimension and can be relevant for the ergodic properties of the system. For hyperbolic maps of the plane with constant Jacobianj, we apply the Bowen-Ruelle formula, using the relationF(=d _H–1)=lnj, which links the Hausdorff dimensiond _H of an attractor to a free energy functionalF() defined in the thermodynamic formalism. We provide numerical evidence that this relation remains valid for some nonhyperbolic maps, such as the Hénon map.     

Strange attractors and asymptotic measures of discrete-time dissipative systems

We investigate asymptotic properties of certain discrete-time dynamical systems in two and three dimensions with solenoidal attractor. It is proved that the asymptotic measures, relevant for the generalized version of the ergodic theorem, all derive from one Haar measure.     

Metrics for ergodicity and design of ergodic dynamics for multi-agent systems

In this paper we propose a metric that quantifies how far trajectories are from being ergodic with respect to a given probability measure. This metric is based on comparing the fraction of time spent by the trajectories in spherical sets to the measure of the spherical sets. This metric is shown to be equivalent to a metric obtained as a distance between a certain delta-like distribution on the trajectories and the desired probability distribution. Using this metric, we formulate centralized feedback control laws for multi-agent systems so that agents trajectories sample a given probability distribution as uniformly as possible. The feedback controls we derive are essentially model predictive controls in the limit as the receding horizon goes to zero and the agents move with constant speed or constant forcing (in the case of second-order dynamics). We numerically analyze the closed-loop dynamics of the multi-agents systems in various scenarios. The algorithm presented in this paper for the design of ergodic dynamics will be referred to as Spectral Multiscale Coverage (SMC).     

Dimension spectrum of axiom a diffeomorphisms. II. Gibbs measures

We compute the dimension spectrumf() of the singularity sets of a Gibbs measure defined on a two-dimensional compact manifold and invariant with respect to aC ² Axiom A diffeomorphism. This case is the generalization of the case where the measure studied is the Bowen-Margulis measure—the one that realizes the topological entropy. We obtain similar results; for example, the functionf is the Legendre-Fenchel transform of a free energy function which is real analytic (linear in the degenerate case). The functionf is also real analytic on its definition domain (defined in one point in the degenerate case) and is related to the Hausdorff dimensions of Gibbs measures singular with respect to each other and whose supports are the singularity sets, and we finally decompose these sets.     

A method for visualization of invariant sets of dynamical systems based on the ergodic partition

We provide an algorithm for visualization of invariant sets of dynamical systems with a smooth invariant measure. The algorithm is based on a constructive proof of the ergodic partition theorem for automorphisms of compact metric spaces. The ergodic partition of a compact metric space A, under the dynamics of a continuous automorphism T, is shown to be the product of measurable partitions of the space induced by the time averages of a set of functions on A. The numerical algorithm consists of computing the time averages of a chosen set of functions and partitioning the phase space into their level sets. The method is applied to the three-dimensional ABC map for which the dynamics was visualized by other methods in Feingold et al. [J. Stat. Phys. 50, 529 (1988)]. (c) 1999 American Institute of Physics.     

Ergodic properties of anomalous diffusion processes

In this paper we study ergodic properties of some classes of anomalous diffusion processes. Using the recently developed measure of dependence called the Correlation Cascade, we derive a generalization of the classical Khinchin theorem. This result allows us to determine ergodic properties of Lévy-driven stochastic processes. Moreover, we analyze the asymptotic behavior of two different fractional Ornstein–Uhlenbeck processes, both originating from subdiffusive dynamics. We show that only one of them is ergodic.     

Generalized Physical and SRB Measures for Hyperbolic Diffeomorphisms

In this paper we introduce the notion of generalized physical and SRB measures. These measures naturally generalize classical physical and SRB measures to measures which are supported on invariant sets that are not necessarily attractors. We then perform a detailed case study of these measures for hyperbolic Hènon maps. For this class of systems we are able to develop a complete theory about the existence, uniqueness, finiteness, and properties of these natural measures. Moreover, we derive a classification for the existence of a measure of full dimension. We also consider general hyperbolic surface diffeomorphisms and discuss possible extensions of, as well as the differences to, the results for Hènon maps. Finally, we study the regular dependence of the dimension of the generalized physical/SRB measure on the diffeomorphism. For the proofs we apply various techniques from smooth ergodic theory including the thermodynamic formalism. 2000 Mathematics Subject Classification. Primary: 37C45, 37D20, 37D35, Secondary: 37A35, 37E30     

On Perturbations of Generalized Landau-Lifshitz Dynamics

We consider deterministic and stochastic perturbations of dynamical systems with conservation laws in ℝ³. The Landau-Lifshitz equation for the magnetization dynamics in ferromagnetics is a special case of our system. The averaging principle is a natural tool in such problems. But bifurcations in the set of invariant measures lead to essential modification in classical averaging. The limiting slow motion in this case, in general, is a stochastic process even if pure deterministic perturbations of a deterministic system are considered. The stochasticity is a result of instabilities in the non-perturbed system as well as of existence of ergodic sets of a positive measure. We effectively describe the limiting slow motion.     

Dimension spectrum of axiom a diffeomorphisms. I. The Bowen-Margulis measure

We compute the dimension spectrumf() of the singularity sets of the Bowen-Margulis measure defined on a two-dimensional compact manifold and invariant with respect to aC ² Axiom A diffeomorphism. It is proved thatf is the Legendre-Fenchel transform of a free energy function which is real analytic (linear in the degenerate case). The functionf is also real analytic on its definition domain (defined in one point in the degenerate case) and is related to the Hausdorff dimensions of Gibbs measures singular with respect to each other and whose supports are the singularity sets, and we decompose these sets.     

On the dimension of deterministic and random Cantor-like sets,symbolic dynamics,and the Eckmann-Ruelle Conjecture

In this paper we unify and extend many of the known results on the dimension of deterministic and random Cantor-like sets in ? ⁿ , and apply these results to study some problems in dynamical systems. In particular, we verify the Eckmann-Ruelle Conjecture for equilibrium measures for Hölder continuous conformal expanding maps and conformal Axiom A^# (topologically hyperbolic) homeomorphims. We also construct a Hölder continuous Axiom A^# homeomorphism of positive topological entropy for which the unique measure of maximal entropy is ergodic and has different upper and lower pointwise dimensions almost everywhere. this example shows that the non-conformal Hölder continuous version of the Eckmann-Ruelle Conjecture is false. The Cantor-like sets we consider are defined by geometric constructions of different types. The vast majority of geometric constructions studied in the literature are generated by a finite collection ofp maps which are either contractions or similarities and are modeled by the full shift onp symbols (or at most a subshift of finite type). In this paper we consider much more general classes of geometric constructions: the placement of the basic sets at each step of the construction can be arbitrary, and they need not be disjoint. Moreover, our constructions are modeled by arbitrary symbolic dynamical systems. The importance of this is to reveal the close and nontrivial relations between the statistical mechanics (and especially the absence of phase transitions) of the symbolic dynamical system underlying the geometric construction and the dimension of its limit set. This has not been previously observed since no phase transitions can occur for subshifts of finite type. We also consider nonstationary constructions, random constructions (determined by an arbitrary ergodic stationary distribution), and combinations of the above.     

Continuity of Eigenfunctions of Uniquely Ergodic Dynamical Systems and Intensity of Bragg Peaks

We study uniquely ergodic dynamical systems over locally compact, sigma-compact Abelian groups. We characterize uniform convergence in Wiener/Wintner type ergodic theorems in terms of continuity of the limit. Our results generalize and unify earlier results of Robinson and Assani respectively. We then turn to diffraction of quasicrystals and show how the Bragg peaks can be calculated via a Wiener/Wintner type result. Combining these results we prove a version of what is sometimes known as the Bombieri/Taylor conjecture. Finally, we discuss various examples including deformed model sets, percolation models, random displacement models, and linearly repetitive systems.     

Recurrence and return times of the Sierpinski carpet

We compute the limit distribution of the recurrence and of the normalizedk th return times to small sets of the Sierpinski carpet with respect to a natural measure defined on it. It is proved that this dynamical system follows the Poisson law, as one could have expected for such schemes. We study different sequences which converge in finite distribution to the Poisson point process. This limit in law is very interesting in ergodic theory, and it seems to appear for chaotic dynamical systems such as the one we study.     

An Ergodic Theorem for the Quantum Relative Entropy

We prove the ergodic version of the quantum Steins lemma which was conjectured by Hiai and Petz. The result provides an operational and statistical interpretation of the quantum relative entropy as a statistical measure of distinguishability, and contains as a special case the quantum version of the Shannon-McMillan theorem for ergodic states. A version of the quantum relative Asymptotic Equipartition Property (AEP) is given.     

Multifractal structure of a riddled basin

For a two-dimensional piecewise linear map with a riddled basin, a multifractal spectrum f(gamma), which characterizes the "skeletons" of the riddled basin, is introduced. With f(gamma), the uncertainty exponent is obtained by a variational principle, which enables us to introduce a concept of a "boundary" for the riddled basin. A conjecture on the relation between f(gamma) and the "stable sets" of various ergodic measures, which coexist with the natural invariant measure on the chaotic attractor, is also proposed. (c) 2001 American Institute of Physics.     

Exponential Convergence for the Stochastically Forced Navier-Stokes Equations and Other Partially Dissipative Dynamics

 We prove that the two dimensional Navier-Stokes equations possess an exponentially attracting invariant measure. This result is in fact the consequence of a more general ``Harris-like' ergodic theorem applicable to many dissipative stochastic PDEs and stochastic processes with memory. A simple iterated map example is also presented to help build intuition and showcase the central ideas in a less encumbered setting. To analyze the iterated map, a general ``Doeblin-like' theorem is proven. One of the main features of this paper is the novel coupling construction used to examine the ergodic theory of the non-Markovian processes. Received: 23 March 2001 / Accepted: 2 April 2002 Published online: 14 October 2002     

Limit Theorems for Self-Similar Tilings

We study deviation of ergodic averages for dynamical systems given by self-similar tilings on the plane and in higher dimensions. The main object of our paper is a special family of finitely-additive measures for our systems. An asymptotic formula is given for ergodic integrals in terms of these finitely-additive measures, and, as a corollary, limit theorems are obtained for dynamical systems given by self-similar tilings.