Ya. B. Pesin

On Rigorous Mathematical Definitions of Correlation Dimension and Generalized Spectrum for Dimensions

We consider different definitions of the correlation dimension and find some relationships between them and other characteristics of dimension type such as Hausdorff dimension, box dimension, etc. We also introduce different ways to define and study the generalized spectrum for dimensions—a one-parameter family of characteristics of dimension type.

Partial Hyperbolicity, Lyapunov Exponents and Stable Ergodicity

We present some results and open problems about stable ergodicity of partially hyperbolic diffeomorphisms with non-zero Lyapunov exponents. The main tool is local ergodicity theory for non-uniformly hyperbolic systems.

Multifractal spectra and multifractal rigidity for horseshoes

We discuss a general concept of multifractality, and give a complete description of the multifractal spectra for Gibbs measures on two-dimensional horseshoes. We discuss a multifractal characterization of surface diffeomorphisms.

Existence and Genericity Problems for Dynamical Systems with Nonzero Lyapunov Exponents

This is a survey-type article whose goal is to review some recent results on existence of hyperbolic dynamical systems with discrete time on compact smooth manifolds and on coexistence of hyperbolic and non-hyperbolic behavior. It also discusses two approaches to the study of genericity of systems with nonzero Lyapunov exponents.

Chaos in Traveling Waves of Lattice Systems of Unbounded Media

We describe coupled map lattices (CMLs) of unbounded media corresponding to some well-known evolution partial differential equations (including reaction-diffusion equations and the Kuramoto-Sivashinsky, Swift-Hohenberg and Ginzburg-Landau equations). Following Kaneko we view CMLs also as phenomenological models of the medium and present the dynamical systems approach to studying the global behavior of solutions of CMLs. In particular, we establish spatio-temporal chaos associated with the set of traveling wave solutions of CMLs as well as describe the dynamics of the evolution operator on this set. Several examples are given to illustrate the appearance of Smale horseshoes and the presence of the dynamics of Morse-Smale type.

On the integrability of intermediate distributions for Anosov diffeomorphisms

We study the integrability of intermediate distributions for Anosov diffeomorphisms and provide an example of a C ∞-Anosov diffeomorphism on a three-dimensional torus whose intermediate stable foliation has leaves that admit only a finite number of derivatives. We also show that this phenomenon is quite abundant. In dimension four or higher this can happen even if the Lyapunov exponents at periodic orbits are constant.