## Thermodynamics of towers of hyperbolic type

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- by Y. Pesin, S. Senti and K. Zhang PDF
- Trans. Amer. Math. Soc.
**368**(2016), 8519-8552 Request permission

## 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 $t_0<0$.## References

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## Additional Information

**Y. Pesin**- Affiliation: Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802
- MR Author ID: 138355
- 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
- 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
- © Copyright 2016 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**368**(2016), 8519-8552 - MSC (2010): Primary 37D25, 37D35, 37E30, 37E35
- DOI: https://doi.org/10.1090/tran/6599
- MathSciNet review: 3551580