Diffeomorphisms with Liao-Pesin set
Abstract
In this paper we mainly deal with an invariant (ergodic) hyperbolic measure μ for a diffeomorphism f, assuming that f is just C1 and for μ a.e. x, the sum of Oseledec spaces corresponding to negative Lyapunov exponents (quasi-limit-)dominates the sum of Oseledec spaces corresponding to positive Lyapunov exponents at x. We generalize a certain of results of Pesin theory from C1+α to the C1 system (f,μ), including a sufficient condition for existence of horseshoe, Livshitz theorem, exponential growth of periodic points, distribution of periodic points, periodic measures, horseshoes, nonuniform specification and lower semi-continuity of entropy function etc. In particular, they are applied for C1 partially hyperbolic systems whose central bundle displays some non-uniform hyperbolicity, including some robust systems. Moreover, for some C1 partially hyperbolic (not necessarily volume-preserving) systems, we get some information of Lebesgue measure on Average-nonuniform hyperbolicityand volume-non-expanding. A constructed machinery is developed for C1 (not necessarily C1+α) diffeomorphisms: new Pesin blocks is established topologically (independent on measures) such that every block has stable manifold theorem and simultaneously has exponential shadowing. The new construction, different with classical C1+α ones, is mainly inspired from Liao's quasi-hyperbolicity and so here we call new blocks by Liao-Pesin blocks and call the new established C1 Pesin theory by C1 Liao-Pesin Theory. Liao-Pesin set not only exists for invariant measures, but also exists for general probability measures, for example, Lebesgue measure (not assuming invariant) in some partially hyperbolic systems.