Thermodynamics of towers of hyperbolic type

Y. Pesin, S. Senti, K. Zhang

Research output: Contribution to journalArticle

13 Scopus citations

Abstract

We introduce a class of continuous maps f of a compact topological space X admitting inducing schemes of hyperbolic type and describe the associated tower constructions. We then establish a thermodynamic formalism, i.e., we describe a class of real-valued potential functions φ on X such that f possesses a unique equilibrium measure μφ, associated to each φ, which minimizes the free energy among the measures that are liftable to the tower. We also describe some ergodic properties of equilibrium measures including decay of correlations and the Central Limit Theorem. We then study the liftability problem and show that under some additional assumptions on the inducing scheme every measure that charges the base of the tower and has sufficiently large entropy is liftable. Our results extend those obtained in previous works of the first and second authors for inducing schemes of expanding types and apply to certain multidimensional maps. Applications include obtaining the thermodynamic formalism for Young’s diffeomorphisms, the Hénon family at the first bifurcation and the Katok map. In particular, we obtain the exponential decay of correlations for equilibrium measures associated to the geometric potentials with t0 < t < 1 for some t0 < 0.

Original languageEnglish (US)
Pages (from-to)8519-8552
Number of pages34
JournalTransactions of the American Mathematical Society
Volume368
Issue number12
DOIs
StatePublished - Jan 1 2016

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

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