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Equilibrium states for hyperbolic potentials via inducing schemes

Jose F. Alves, Krerley Oliveira, Eduardo Santana

Abstract

In a context of non-uniformly expanding maps, possibly with the presence of a critical set, we prove the existence of finitely many ergodic equilibrium states for hyperbolic potentials. Moreover, the equilibrium states are expanding measures. This generalizes a result due to Ramos and Viana, where analytical methods are used for maps with no critical sets. The strategy here consists in using a finite number of inducing schemes with a Markov structure in infinitely many symbols to code the dynamics, to obtain an equilibrium state for the associated symbolic dynamics and then projecting it to obtain an equilibrium state for the original map. We apply our results to the important class of multidimensional Viana maps.

Equilibrium states for hyperbolic potentials via inducing schemes

Abstract

In a context of non-uniformly expanding maps, possibly with the presence of a critical set, we prove the existence of finitely many ergodic equilibrium states for hyperbolic potentials. Moreover, the equilibrium states are expanding measures. This generalizes a result due to Ramos and Viana, where analytical methods are used for maps with no critical sets. The strategy here consists in using a finite number of inducing schemes with a Markov structure in infinitely many symbols to code the dynamics, to obtain an equilibrium state for the associated symbolic dynamics and then projecting it to obtain an equilibrium state for the original map. We apply our results to the important class of multidimensional Viana maps.

Paper Structure

This paper contains 15 sections, 29 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Topological pressure
  3. Expanding measures and hyperbolic potentials
  4. Expanding potentials
  5. Viana maps
  6. Equilibrium states for lifted dynamics
  7. Inducing schemes
  8. Hyperbolic potentials
  9. Thermodynamic formalism on inducing schemes
  10. Zero pressure and equilibrium states
  11. Finiteness of ergodic equilibrium states
  12. Measures with low free energy
  13. Integrability of the inducing time
  14. Hyperbolic and Expanding Potentials
  15. Acknowledgement

Key Result

Theorem A

Let $f:M \to M$ be a continuous map and $\phi:M \to \mathbb{R}$ be a Hölder continuous potential with $P_{f}(\phi)<\infty$. If there exist $0<\sigma<1$ and $\varepsilon>0$ such that $\phi$ is a $(\sigma,\varepsilon)$-hyperbolic potential, then there exists some ergodic equilibrium state for $\phi$.

Theorems & Definitions (48)

  • Theorem A
  • Theorem B
  • Theorem C
  • Proposition 2.1: Zweimüller Z
  • Theorem 2.2: Pinheiro Pi1
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Corollary 2.5
  • proof
  • ...and 38 more