Showing papers by "Lai Sang Young published in 1998"

Statistical properties of dynamical systems with some hyperbolicity

Abstract: This paper is about the ergodic theory of attractors and conservative dynamical systems with hyperbolic properties on large parts (though not necessarily all) of their phase spaces. The main results are for discrete time systems. To put this work into context, recall that for Axiom A attractors the picture has been fairly complete since the 1970’s (see [S1], [B], [R2]). Since then much progress has been made on two fronts: there is a general nonuniform theory that deals with properties common to all diffeomorphisms with nonzero Lyapunov exponents ([O], [P1], [Ka], [LY]), and there are detailed analyses of specific kinds of dynamical systems including, for example, billiards, 1-dimensional and Henon-type maps ([S2], [BSC]; [HK], [J]; [BC2], [BY1]). Statistical properties such as exponential decay of correlations are not enjoyed by all diffeomorphisms with nonzero Lyapunov exponents. The goal of this paper is a systematic understanding of these and other properties for a class of dynamical systems larger than Axiom A. This class will not be defined explicitly, but it includes some of the much studied examples. By looking at regular returns to sets with good hyperbolic properties, one could give systems in this class a simple dynamical representation. Conditions for the existence of natural invariant measures, exponential mixing and central limit theorems are given in terms of the return times. These conditions can be checked in concrete situations, giving a unified way of proving a number of results, some new and some old. Among the new results are the exponential decay of correlations for a class of scattering billiards and for a positive measure set of Henon-type maps.

Computing invariant measures for expanding circle maps

Abstract: Let f be a sufficiently expanding circle map. We prove that a certain Markov approximation scheme based on a partition of into equal intervals produces a probability measure whose total variation norm distance from the exact absolutely continuous invariant measure is bounded by ; C is a constant depending only on the map f.

Developments in chaotic dynamics

Abstract: 1318 NOTICES OF THE AMS VOLUME 45, NUMBER 10 Dynamical systems as a mathematical discipline goes back to Poincaré, who developed a qualitative approach to problems that arose from celestial mechanics. The subject has expanded considerably in scope and has undergone some fundamental changes in the last three decades. Today it stands at the crossroads of several areas of mathematics, including analysis, geometry, topology, probability, and mathematical physics. It is generally regarded as a study of iterations of maps, of time evolutions of differential equations, and of group actions on manifolds. This article is about an area of dynamical systems called hyperbolic dynamics or chaotic dynamics. The concept of hyperbolicity, which we will define shortly, was used by Hedlund and Hopf in their analysis of geodesic flows on manifolds with negative curvature. A systematic study of hyperbolic systems began in the 1960s when Smale outlined in his 1967 Bulletin of the AMS article [Sm] a program for the geometric theory of dynamical systems. Another viewpoint, namely, the ergodic theory or probability approach to hyperbolic dynamics, was introduced several years later by Sinai and Ruelle. These ideas have developed over the last thirty years into a very rich theory, one that has changed the qualitative theory of ordinary differential equations and helped shape modern ideas about chaos. In this article I would like to report on some developments since the 1960s. This, however, is very far from a survey. I hope that by focusing on a couple of examples and a small sample of ideas I can convey to the general mathematics community a sense of some of the progress that has been made.