Diffeomorphisms with Liao-Pesin set

In this paper we mainly deal with an invariant (ergodic) hyperbolic measure $\mu$ for a diffeomorphism $f,$ assuming that $f$ is just $C^1$ and for $\mu$ 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 $C^{1+\alpha}$ to the $C^1$ system $ (f,\mu)$, 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 $C^1$ partially hyperbolic systems whose central bundle displays some non-uniform hyperbolicity, including some robust systems. Moreover, for some $C^1$ 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 $C^1$ (not necessarily $C^{1+\alpha}$) 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 $C^{1+\alpha}$ 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 $C^1$ Pesin theory by $C^1$ 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.