# 12月12日 动力系统系列学术报告（数学与统计学院）

For partially hyperbolic diffeomorphism with mostly expanding and mostly contracting centers, we establish a topological structure, called skeleton. We build the one-to-one corresponding between periodic points in any skeleton and physical measures. By making perturbations on skeletons, we study the continuity of physical measures with respect to dynamics under C^1-topology.

Let $f$ be a holomorphic endomorphism on $\mathbb{P}^2$. The first Julia set  $J_1$ is classically defined as the maximal locus such that $\left\{f^n\right\}$ locally do not form a normal family. The second Julia set $J_2\subset J_1$ is defined as the support of the measure of maximal entropy. In this talk we will study these two Julia sets for post-critically finite (PCF for short) maps. Here are two main results: 1. $J_1\setminus J_2$ is contained in the union of attracting basins of critical component cycles and stable manifolds of sporadic super-saddle points. 2. If $x\in J_2$ is not contained in the stable manifold of a sporadic super-saddle point, then there is no Fatou disk containing $x$. As corollaries of our results, 1. We answer some questions of Fornaess-Sibony about the non-wandering set for PCF maps. 2. We give a new proof of de Thelin’s laminarity of the Green current on $J_1\setminus J_2$. 3. We obtain characterizations of PCF maps which are expanding on $J_2$ or satisfy Axiom A.

Let f be a non-uniformly hyperbolic system with positive entropy,  and let A be a Holder continuous cocycle of injective bounded linear operators acting on a Banach space.  We prove that  there is a sequence of horseshoes  for f and  dominated splittings for A on the horseshoes, such that not only the  measure theoretic entropy  of f but also the Lyapunov exponents of  A   can be approximated by the topological entropy of f and the Lyapunov exponents of A on the horseshoes, respectively.

We consider a smooth diffeomorphism on a compact manifold. Ledrappier and Young introduced entropies along unstable foliations for an ergodic measure. We relate those entropies to covering numbers in order to give a new upper bound on the metric entropy in terms of Lyapunov exponents and topological entropy or volume growth of sub-manifolds.

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