Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Equilibrium Stability for Open Zooming Systems

Rafael A. Bilbao, Eduardo Santana

TL;DR

The paper addresses equilibrium stability for open zooming systems by developing a unified framework of zooming sets, holes, and zooming potentials, and proving that equilibrium states depend continuously on the dynamics and potential. The authors introduce finite Markov structures adapted to holes, define open pressure and zooming potentials, and establish finiteness and stability of equilibrium states, including in skew-product settings via the Ledrappier–Walters decomposition. They show that the equilibrium states for a broad class of zooming potentials are not only finite but vary continuously under perturbations, and they prove uniqueness for continuous zooming potentials using stability and density arguments. The results are illustrated across diverse non-uniformly expanding models (Viana, Benedicks–Carleson, Rovella) and their open-system variants, highlighting robustness of the statistical description and enabling applications to a wide range of systems with holes and non-exponential contractions.

Abstract

We prove that for a wide family of open zooming systems and zooming potentials we have equilibrium stability, i.e., the equilibrium states depend continuously on the dynamics and the potential. We consider the open zooming systems with special holes and quite general contractions and zooming potentials with locally Hölder induced potential, which include the Hölder ones. We also prove stability for skew-products with the base being a zooming system like above. As a consequence of finiteness and stability, we obtain uniqueness of equilibrium state.

Equilibrium Stability for Open Zooming Systems

TL;DR

The paper addresses equilibrium stability for open zooming systems by developing a unified framework of zooming sets, holes, and zooming potentials, and proving that equilibrium states depend continuously on the dynamics and potential. The authors introduce finite Markov structures adapted to holes, define open pressure and zooming potentials, and establish finiteness and stability of equilibrium states, including in skew-product settings via the Ledrappier–Walters decomposition. They show that the equilibrium states for a broad class of zooming potentials are not only finite but vary continuously under perturbations, and they prove uniqueness for continuous zooming potentials using stability and density arguments. The results are illustrated across diverse non-uniformly expanding models (Viana, Benedicks–Carleson, Rovella) and their open-system variants, highlighting robustness of the statistical description and enabling applications to a wide range of systems with holes and non-exponential contractions.

Abstract

We prove that for a wide family of open zooming systems and zooming potentials we have equilibrium stability, i.e., the equilibrium states depend continuously on the dynamics and the potential. We consider the open zooming systems with special holes and quite general contractions and zooming potentials with locally Hölder induced potential, which include the Hölder ones. We also prove stability for skew-products with the base being a zooming system like above. As a consequence of finiteness and stability, we obtain uniqueness of equilibrium state.

Paper Structure

This paper contains 23 sections, 20 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Acknowledgement
  3. Setup and Main Results
  4. Zooming Sets and Measures
  5. Markov structure with special holes
  6. Pressure, zooming potentials and equilibrium states
  7. Equilibrium stability
  8. Skew products
  9. Proof of Theorem \ref{['A']}
  10. Proof of Theorem \ref{['thB']}
  11. Proof of Theorem \ref{['C']}
  12. Applications
  13. Viana maps
  14. Benedicks-Carleson Maps
  15. Rovella Maps
  16. ...and 8 more sections

Key Result

Theorem 2.2.1

Let $(f,M,\mu,\Lambda)$ be a zooming system. Let $r_{0} > 0$ and let $\mathcal{A} = \{B_{r}(p_{i}) \mid i=1,2,\dots,k , r < r_{0}\}$ be a finite open cover of $M$ such that $B_{r}(p_{i}) \cap B_{r/2}(p_{j}) = \emptyset, \text{for all} \, \, i,j \leq k, j\neq i$. We assume that $H \subset M$ satisfie for some $k_{1},\dots,k_{t} \leq k$. Then if $r_{0}$ is sufficiently small, depending only on $(f,M

Theorems & Definitions (46)

  • Definition 2.1.1
  • Definition 2.1.2
  • Definition 2.1.3
  • Definition 2.1.4
  • Remark 2.1.1
  • Definition 2.1.5
  • Theorem 2.2.1
  • Definition 2.3.1
  • Definition 2.3.2
  • Definition 2.3.3
  • ...and 36 more