Yakov Pesin

Lectures on partial hyperbolicity and stable ergodicity

Weak integrability of the central foliation 56 6. Intermediate Foliations 58 6.1. Non-integrability of intermediate distributions 58 6.2. Invariant families of local manifolds 59 6.3. Insufficient smoothness of intermediate foliations 64 7. Absolute Continuity 69 7.1. The holonomy map 69 7.2. Absolute continuity of local manifolds 75

The multifractal analysis of Gibbs measures: Motivation, mathematical foundation, and examples

We first motivate the study of multifractals. We then present a rigorous mathematical foundation for the multifractal analysis of Gibbs measures invariant under dynamical systems. Finally we effect a complete multifractal analysis for several classes of hyperbolic dynamical systems. (c) 1997 American Institute of Physics.

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 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 ℝn, 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.

EQUILIBRIUM MEASURES FOR MAPS WITH INDUCING SCHEMES

We introducea class of continuousmaps f of a compact topologi- cal space I admitting inducing schemes and describe the tower constructions associated with them. We then establish a thermodynamicformalism, i.e., de- scribe a class of real-valued potential functions ϕ on I, which admit a unique equilibrium measure ¹ ϕ minimizing the free energy for a certain class of in- variant measures. We also describe ergodic properties of equilibrium mea- sures, including decay of correlation and the Central Limit Theorem. Our re- sults apply to certain maps of the interval with critical points and/or singular- ities (including some unimodal and multimodal maps) and to potential func- tions ϕt =−t log|d f | with t ∈(t0,t1) for some t0 < 1< t1. In theparticularcase of S-unimodal maps we show that one can choose t0 < 0 and that the class of measures under consideration consists of all invariant Borel probability mea- sures.

An analytical approach to the problem of inverse optimization with additive objective functions: an application to human prehension

We consider the problem of what is being optimized in human actions with respect to various aspects of human movements and different motor tasks. From the mathematical point of view this problem consists of finding an unknown objective function given the values at which it reaches its minimum. This problem is called the inverse optimization problem. Until now the main approach to this problems has been the cut-and-try method, which consists of introducing an objective function and checking how it reflects the experimental data. Using this approach, different objective functions have been proposed for the same motor action. In the current paper we focus on inverse optimization problems with additive objective functions and linear constraints. Such problems are typical in human movement science. The problem of muscle (or finger) force sharing is an example. For such problems we obtain sufficient conditions for uniqueness and propose a method for determining the objective functions. To illustrate our method we analyze the problem of force sharing among the fingers in a grasping task. We estimate the objective function from the experimental data and show that it can predict the force-sharing pattern for a vast range of external forces and torques applied to the grasped object. The resulting objective function is quadratic with essentially non-zero linear terms.

Chapter 2 – Smooth Ergodic Theory and Nonuniformly Hyperbolic Dynamics

This chapter discusses smooth ergodic theory and nonuniformly hyperbolic dynamics. Smooth ergodic theory studies topological and ergodic properties of smooth dynamical systems with nonzero Lyapunov exponents. There are two classes of hyperbolic invariant measures on compact manifolds for which one can obtain a sufficiently complete description of its ergodic properties. They are: smooth measures, that is, measures that are equivalent to the Riemannian volume with the Radon-Nikodym derivative bounded from above and bounded away from zero, and Sinai-Ruelle-Bowen measures. Nonuniform hyperbolicity conditions can be expressed in terms of the Lyapunov exponents—that is, a dynamical system is nonuniformly hyperbolic if it admits an invariant measure with nonzero Lyapunov exponents almost everywhere. This provides an efficient tool in verifying the nonuniform hyperbolicity conditions and determines the importance of the nonuniform hyperbolicity theory in applications. The nonuniform hyperbolicity theory covers an enormous area of dynamics, such as nonuniformly hyperbolic one-dimensional transformations, random dynamical systems with nonzero Lyapunov exponents, billiards and related systems (for example, systems of hard balls), and numerical computation of Lyapunov exponents.

Introduction to Smooth Ergodic Theory

Table of contents: The core of the theory: * Examples of hyperbolic dynamical systems * General theory of Lyapunov exponents * Lyapunov stability theory of nonautonomous equations * Elements of the nonuniform hyperbolicity theory * Cocycles over dynamical systems * The Multiplicative Ergodic Theorem * Local manifold theory * Absolute continuity of local manifolds * Ergodic properties of smooth hyperbolic measures * Geodesic flows on surfaces of nonpositive curvature Selected advanced topics: * Cone technics * Partially hyperbolic diffeomorphisms with nonzero exponents * More examples of dynamical systems with nonzero Lyapunov exponents * Anosov rigidity * C^1 pathological behavior: Pughs example * Bibliography * Index

Dimension of non-conformal repellers: a survey

This paper is a survey of recent results on the dimension of repellers for expanding maps and limit sets for iterated function systems. While the case of conformal repellers is well understood, the study of non-conformal repellers is in its early stages though a number of interesting phenomena have been discovered, some remarkable results obtained and several interesting examples constructed. We will describe contemporary state of the art in the area with emphasis on some new emerging ideas and open problems.

Thermodynamics of towers of hyperbolic type

We introduce a class of continuous maps