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Regularity, linear response formula and differentiability of the free energy for non-uniformly expanding local homeomorphisms

Carlos Bocker, Ricardo Bortolotti, Armando Castro, Sávio Santana

Abstract

We study equilibrium states for an open class of non-uniformly expanding local homeomorphisms defined by a mild condition such that for some iterate each point admits at least one contracting inverse branch. We prove the existence and uniqueness of equilibrium states and the differentiability of statistical quantities (such as the equilibrium states and the free energy function) with respect to the dynamical system.

Regularity, linear response formula and differentiability of the free energy for non-uniformly expanding local homeomorphisms

Abstract

We study equilibrium states for an open class of non-uniformly expanding local homeomorphisms defined by a mild condition such that for some iterate each point admits at least one contracting inverse branch. We prove the existence and uniqueness of equilibrium states and the differentiability of statistical quantities (such as the equilibrium states and the free energy function) with respect to the dynamical system.

Paper Structure

This paper contains 16 sections, 10 theorems, 39 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Statements of the results
  3. Hölder continuous setting
  4. $C^{r+\alpha}$-smooth setting
  5. Statistical consequences
  6. Central Limit Theorem
  7. Free energy
  8. Legendre transform and large deviations principle
  9. The Main Proposition
  10. Proof of the Theorems
  11. Examples
  12. Bad set covered with few domains of injectivity (as in BCV16)
  13. Derived from expanding
  14. Manneville-Pomeau like maps
  15. Hyperbolic potentials for topologically exact local homeomorphism
  16. ...and 1 more sections

Key Result

Theorem 1

(Equilibrium states for Hölder potentials) Given $\gamma\in(0,1)$ and integers $N, G \geq 2$, there are $L>1$ and $\varepsilon>0$ such that for all $f\in {\mathcal{H}}^\alpha_{\gamma, L, N, G}$ and for all potential $\phi\in P^\alpha(\varepsilon)$ the transfer operator ${\mathcal{L}}_{f,\phi}$ has t Moreover, the following maps are continuous: And they are analytic with respect to $\phi$.

Figures (1)

  • Figure 1: Graph of Maneville-Pomeau like map $f$.

Theorems & Definitions (21)

  • Theorem 1
  • Remark
  • Theorem 2
  • Corollary 3: Central Limit Theorem
  • Corollary 4: Differentiability of the Free Energy
  • Remark 2.1
  • Corollary 5: Large Deviation Principle with Differentiable Rate
  • Proposition 3.1: Main Proposition
  • Lemma 3.2
  • proof
  • ...and 11 more