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Mathematische Zeitschrift

, Volume 291, Issue 3–4, pp 1451–1494 | Cite as

Pesin’s entropy formula for \(C^1\) non-uniformly expanding maps

  • Vítor AraujoEmail author
  • Felipe Santos
Article
  • 28 Downloads

Abstract

We prove existence of equilibrium states with special properties for a class of distance expanding local homeomorphisms on compact metric spaces and continuous potentials. Moreover, we formulate a C\(^1\) generalization of Pesin’s Entropy Formula: all ergodic weak-SRB-like measures satisfy Pesin’s Entropy Formula for \(C^1\) non-uniformly expanding maps. We show that for weak-expanding maps such that \({\text {Leb}}\)-a.e x has positive frequency of hyperbolic times, then all the necessarily existing ergodic weak-SRB-like measures satisfy Pesin’s Entropy Formula and are equilibrium states for the potential \(\psi =-\log |\det Df|\). In particular, this holds for any \(C^1\)-expanding map and, in this case, the set of invariant probability measures that satisfy Pesin’s Entropy Formula is the weak\(^*\)-closed convex hull of the ergodic weak-SRB-like measures.

Keywords

Non-uniform expansion SRB/physical-like measures Equilibrium states Pesin’s entropy formula \(C^1\) smooth Uniform expansion 

Mathematics Subject Classification

Primary 37D25 Secondary 37D35 37D20 37C40 

Notes

Acknowledgements

This is the PhD thesis of F. Santos at the Instituto de Matematica e Estatistica-Universidade Federal da Bahia (UFBA) under a CAPES scholarship. He thanks the Mathematics and Statistics Institute at UFBA for the use of its facilities and the finantial support from CAPES during his M.Sc. and Ph.D. studies. The authors thank the anonymous referee for the careful reading and the many useful suggestions that greatly helped to improve the quality of the text.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Instituto de MatemáticaUniversidade Federal da Bahia Av. Ademar de Barros s/nSalvadorBrazil
  2. 2.Centro de Formação de ProfessoresUniversidade Federal do Reconcavo da Bahia, Avenida NestorAmargosaBrazil

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