Unstable entropy of partially hyperbolic diffeomorphisms along non-compact subsets. (30th May 2019)

• Record Type: Journal Article
• Authors: Ponce, Gabriel
• Is Part Of: Nonlinearity. Volume 32:Number 7(2019)
• Page Start: 2337
• Page End: 2351
• Publication Date: 2019-05-30
• Subjects: unstable entropy -- partially hyperbolic dynamics -- entropy of non-compact subsets; 37D30 -- 37A35; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/ab1ba3
• Languages: English
• ISSNs: 0951-7715

Abstract

Abstract: Given a partially hyperbolic diffeomorphism defined on a compact Riemannian manifold M, in this paper we define the concept of unstable topological entropy of f on a set not necessarily compact. Using recent results of Yang (2016 (arXiv:1601.05504)) and Hu et al (2017 Adv. Math .321 31–68) we extend a theorem of Bowen (1973 Trans. Am. Math. Soc .184 125–36) proving that, for an ergodic f -invariant measure, the unstable measure theoretical entropy of f is upper bounded by the unstable topological entropy of f on any set of positive -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 Hu et al (2017 Adv. Math .321 31–68).