Product Structure of Hyperbolic Measures

Abstract

We describe in this chapter the almost product structure of the hyperbolic invariant measures, that is, the invariant measures with nonzero Lyapunov exponents almost everywhere. We note that the existence of a hyperbolic measure ensures the presence of nonuniform hyperbolicity almost, everywhere, which together with the nontrivial recurrence given by the invariant measure causes a very complicated behavior of the system. It turns out that, to some extent, the almost product structure of a hyperbolic measure imitates the local product structure defined by the local stable and unstable manifolds, but its study is much more delicate. We also describe the relation between the product structure of hyperbolic invariant measures and the dimension theory of dynamical systems.