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Thermodynamics of towers of hyperbolic type


Authors: Y. Pesin, S. Senti and K. Zhang
Journal: Trans. Amer. Math. Soc. 368 (2016), 8519-8552
MSC (2010): Primary 37D25, 37D35, 37E30, 37E35
DOI: https://doi.org/10.1090/tran/6599
Published electronically: March 1, 2016
MathSciNet review: 3551580
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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 $ \varphi $ on $ X$ such that $ f$ possesses a unique equilibrium measure $ \mu _\varphi $, associated to each $ \varphi $, 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 $ t_0< t<1$ for some $ t_0<0$.


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Additional Information

Y. Pesin
Affiliation: Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802
Email: pesin@math.psu.edu

S. Senti
Affiliation: Instituto de Matematica, Universidade Federal do Rio de Janeiro, C.P. 68 530, CEP 21941-909, Rio de Janeiro, Brazil
Email: senti@im.ufrj.br

K. Zhang
Affiliation: Department of Mathematics, University of Toronto, Toronto, Ontario, Canada
Email: kzhang@math.utoronto.ca

DOI: https://doi.org/10.1090/tran/6599
Received by editor(s): March 12, 2014
Received by editor(s) in revised form: October 13, 2014
Published electronically: March 1, 2016
Additional Notes: The first author was partially supported by the National Science Foundation grant #DMS-1400027. The second author acknowledges the support of the CNPq, CAPES, PRONEX and BREUDS
Article copyright: © Copyright 2016 American Mathematical Society