Variational equalities of entropy in nonuniformly hyperbolic systems

Abstract

Abstract. In this paper we prove that for an ergodic hyperbolic measure ω of a C 1+α diffeomorphism f on a Riemannian manifold M , there is an ω-full measured set Λ such that for every invariant probability µ ∈ M inv ( Λ, f ), the metric entropy of µ is equal to the topological entropy of saturated set Gµ consisting of generic points of µ:hµ(f ) = htop(f, Gµ).

Topological entropy of level sets of empirical measures for non-uniformly expanding maps

“…To finalize the proof of Theorem 2.1 (1) we are left to give the estimate on the topological entropy of the set QG µ1,µ2 ( ). Using (20), (17) and…”

“…In particular, this provides a criterium for creating Birkhoff irregular points using orbits of points which are generic for different ergodic measures, something that was not possible using [19,31]. This consists of a method different from the improved shadowing lemma developed by C. Liang, G. Liao, W. Sun and X. Tian [17] in the context of non-uniformly hyperbolic diffeomorphisms. Our main result (Theorem 2.1) is that, under (topological) non-uniform specification properties, the topological entropy of the set of points whose empirical measures accumulates on two ergodic measures is bounded below by the minimum entropy among both measures.…”

Multifractal Analysis on the Divergence Set of Asymptotically Additive Sequence

“…Takens considered divergence point further more in [18] and proposed the following problem: whether there are persistent classes of smooth dynamical systems such that the set initial states which give rise to orbits with historic behaviour has "positive Lebesgue measure". This problem is called "Takens last problem", see [12,13,14,15].…”

Upper capacity entropy and packing entropy of saturated sets for amenable group actions

“…The g-almost product property is weaker than specification and uniform separation is weaker than expansiveness. Also the equation 1.1 is generalized into non-uniformly hyperbolic systems in [13] and non-uniformly expanding maps in [20]. Besides Bowen topological entropy, packing topological entropy and upper capacity topological entropy are another two important concepts for characterizing the size of non-compact subset Z ⊂ X.…”