This project is devoted to the study of ergodic properties of nonuniformly hyperbolic systems. We focus on almost hyperbolic systems and some related systems. An smooth dynamical system is almost hyperbolic if it is hyperbolic everywhere except at a finite set of points. Such systems may have quite different ergodic behaviors from uniformly hyperbolic systems. In this project we will study existence of equilibrium states, including SRB measures and absolutely continuous invariant measures in multidimensional spaces; rates of convergence to the equilibrium states for both finite and infinite measure cases, and some related topics such as rates of decay of correlations of the systems and the central limit theorem; some other ergodic properties of the systems such as stochastic stability, Gibbs properties, topological conjugation. We are also interested in using these or similar systems to construct varies examples of systems that have given properties, for instance, diffeomorphisms or flows on any manifolds that preserve the Riemannian volume, have nonzero Lyapunov exponents almost everywhere, and have countably many ergodic components.
Ergodic theory concerns the statistic behavior of systems. Ergodic properties of uniformly hyperbolic systems were the main research subject in smooth dynamical systems from 60's to 80's. The behaviors of such systems are regarded as chaotic. Now nonuniformly hyperbolic systems become a main research topic in the field. This project is devoted to the study of ergodic properties of almost hyperbolic systems and some other related systems. Almost hyperbolic systems are smooth dynamical systems in which hyperbolic conditions are violated at only finitely number of points. These systems lie on the boundary of the set of uniformly hyperbolic systems, and are the simplest but nontrivial nonuniformly hyperbolic systems. Earlier studies on examples of such systems, such as systems on the intervals and torus, show that some ergodic properties may change dramatically. In this project we try to develop some theorems for general almost hyperbolic systems rather than individual examples.
