Statistical properties of some almost hyperbolic systems. Smooth Ergodic Theory and its Applications

Abstract. Almost hyperbolic systems are smooth dynamical systems that are hyperbolic everywhere except for a nite set of points. Evidences from low-dimensional cases show that the topological properties of such systems are similar to that of uniformly hyperbolic systems, and the ergodic properties may be quite dierent. It admit SRB measures or innite SRB measures. In the latter case, the systems are statistically deterministic in the sense that almost every orbit spends one hundred percent of its time arbitrarily close to the indierent xed points, though they are still topologically mixing. Even in the former case, correlation decay may change from exponential to power law. A smooth dynamical system is almost hyperbolic if it is hyperbolic everywhere except at a nite set of points. Ergodic properties of an almost hyperbolic system may be quite dierent from that of a uniformly hyperbolic system, even though we can make them topologically conjugate