Erodic Properties for 'Almost Hyperbolic' Systems

Abstract

Hu

This proposed research addresses ergodic properties of 'almost hyperbolic' dynamical systems.

Here 'almost hyperbolic' systems means a smooth dynamical systems that is hyperbolic everywhere except for a finite set of points. These systems include 'almost Anosov' diffeomorphisms, 'almost hyperbolic' invariant sets, piecewise expanding maps on the unit interval with indifferent fixed points, and invariant sets of expanding maps with indifferent fixed points. 'Almost hyperbolic' systems lie on the boundary of the set of uniformly hyperbolic systems in the space of smooth dynamical systems. By the results obtained from 'almost Anosov' diffeomorphisms on the surface, the ergodic properties of such systems may be quite different from that of uniformly hyperbolic systems, though the topological properties are similar. For example, these systems may or may not admit SRB measures, and even when they do, correlation decay may change from exponential to polynomial.

This project stresses existence and properties of SRB measures, rate of decay of correlations, and stochastic stability under small perturbations for general 'almost hyperbolic' systems.

Uniformly hyperbolic systems are the main research subjects in smooth dynamical systems since late 60's. These systems display many types of complex dynamic behavior, and whose behavoir are often regarded as chaotic. Ergodic theory concerns understanding the long-term behavior of systems. Now ergodic properties for uniformly hyperbolic systems are understood very well, and people are interested in such properties for nonuniformly hyperbolic systems. 'Almost hyperbolic' means that hyperbolic conditions fail at a finite set of points. These systems lie on the boundary of the set of uniformly hyperbolic systems, and are the simplest nonuniformly hyperbolic systems. Results obtained earlier for some particular systems, 'almost Anosov' systems in the two-dimensional torus, show that such systems may-and sometimes do-exhibit totally different long term behavior. In this project we will study ergodic properties of more general 'almost hyperbolic' systems. We are particular interested in orbit distrubitions, rate of mixing, and stochastic stability under small perturbation for such systems.