Unstable entropy of partially hyperbolic diffeomorphisms along non-compact subsets

Abstract

Given a partially hyperbolic diffeomorphism f : M → M defined on a compact Riemannian manifold M , in this paper we define the concept of unstable topological entropy of f on a set Y ⊂ M not necessarily compact. Using recent results of J. Yang [17] and H. Hu, Y. Hua and W. Wu [5] we extend a theorem of R. Bowen [2] proving that, for an ergodic f -invariant measure µ, the unstable measure theoretical entropy of f is upper bounded by the unstable topological entropy of f on any set of positive µ-measure. We define a notion of unstable topological entropy of f using a Hausdorff dimension like characterization and we prove that this definition coincides with the definition of unstable topological entropy introduced in [5].

Unstable entropies and dimension theory of partially hyperbolic systems

“…When the paper was being written we found that recently Ponce [31] used Bowen's original ideas [8] to define unstable topological entropy of subsets and get corollaries A.1 and A.2 for C 1 -PHDs. Here we use the methods of Caratheódory-Pesin dimension to define unstable topological entropy of subsets and develop more general results.…”

Unstable entropy in smooth ergodic theory *

“…Entropy of non-compact sets and Carathéodory dimension. Similar to Bowen approach, Ponce [55] and independently Tian-Wu [69] have developed the notion of unstable entropy for non-compact subsets. As these de nitions are in the framework of Carathéodory dimension, we refer to works of Pesin-Pitskel [52] and the reference book of Pesin [53] for more general de nition of Carathéodory construction.…”

Unstable pressure and u-equilibrium states for partially hyperbolic diffeomorphisms

“…Naturally, one can define the unstable pressure via Caratheódory structure following [13]. For the unstable entropy case, this has been done in [15,19].…”