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Weak Gibbs Measures for Local Homeomorphisms

Giovane Ferreira, Vanessa Ramos

Abstract

We study a broad class of local homeomorphisms and continuous potentials, proving the existence and uniqueness of weak Gibbs measures. From the Gibbs property, we show the uniqueness of equilibrium states and derive a large deviations principle. Furthermore, we extend these results to a class of attractors that are semiconjugate to local homeomorphisms from our original setting. Our approach is based on non-uniform conformal-like property and applies to a wide range of topological dynamical systems, including non-uniformly expanding maps, zooming local homeomorphisms, attractors arising from solenoid-like constructions and certain families of partially hyperbolic horseshoes.

Weak Gibbs Measures for Local Homeomorphisms

Abstract

We study a broad class of local homeomorphisms and continuous potentials, proving the existence and uniqueness of weak Gibbs measures. From the Gibbs property, we show the uniqueness of equilibrium states and derive a large deviations principle. Furthermore, we extend these results to a class of attractors that are semiconjugate to local homeomorphisms from our original setting. Our approach is based on non-uniform conformal-like property and applies to a wide range of topological dynamical systems, including non-uniformly expanding maps, zooming local homeomorphisms, attractors arising from solenoid-like constructions and certain families of partially hyperbolic horseshoes.
Paper Structure (17 sections, 28 theorems, 181 equations, 1 figure)

This paper contains 17 sections, 28 theorems, 181 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Statement of results
  3. Examples
  4. Topological Pressure
  5. The Reference measure
  6. Weak Gibbs measure
  7. Absolutely continuous invariant measure
  8. Equilibrium state for local homeomorphisms.
  9. Liftability of the weak Gibbs measure
  10. Cohomologous potential
  11. Gibbs measure for the attractor
  12. Equilibrium states for the attractor.
  13. Large Deviation Results
  14. Upper bound
  15. Lower bound over ergodic measures
  16. ...and 2 more sections

Key Result

Theorem A

Consider $f:M\to M$ a local homeomorphism satisfying (H1) and ${\varphi}: M \to \mathbb{R}$ a continuous potential satisfying (H2) and (H3). Then there exists a unique ergodic weak Gibbs measure $\mu_{\varphi}$ for $(f, {\varphi})$. Moreover, the relative topological pressure of $\mathcal{G}$ is equ

Figures (1)

  • Figure 1: Deformations satisfying Example \ref{['deformacao']}.

Theorems & Definitions (57)

  • Remark 2.1
  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Theorem E
  • Theorem F
  • Example 3.1
  • Lemma 3.1
  • Example 3.2
  • ...and 47 more