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Quantitative statistical stability for the equilibrium states of piecewise partially hyperbolic maps

Rafael Bilbao, Ricardo Bioni, Rafael Lucena

TL;DR

The paper addresses how equilibrium states for a class of skew-product maps $F=(f,G)$, with a non-uniformly expanding base and a contracting fiber, respond to deterministic perturbations. It develops an anisotropic space $S^\infty$ for disintegrated measures and harnesses Hölder regularity of disintegration to obtain a spectral-gap framework for the transfer operator $\mathrm{F}_*$, enabling quantitative statistical stability. The authors prove existence and uniqueness of the invariant measure in $S^\infty$ and establish exponential convergence to equilibrium, then formulate and apply a perturbation theory via an $(R({\cdot}), {\cdot})$-family of operators to obtain a modulus of continuity $O(\delta^{\zeta}\log \delta)$ for the perturbed invariant measures. This yields precise, rate-based stability results for a broad class of deterministic perturbations, contributing to the understanding of uncertainty propagation in complex non-invertible partially hyperbolic systems.

Abstract

We consider a class of endomorphisms that contains a set of piecewise partially hyperbolic dynamics semi-conjugated to non-uniformly expanding maps. Our goal is to study a class of endomorphisms that preserve a foliation that is almost everywhere uniformly contracted, with possible discontinuity sets parallel to the contracting direction. We apply the spectral gap property and the $ζ$-Hölder regularity of the disintegration of its equilibrium states to prove a quantitative statistical stability statement. More precisely, under deterministic perturbations of the system of size $δ$, we show that the $F$-invariant measure varies continuously with respect to a suitable anisotropic norm. Moreover, we prove that for certain interesting classes of perturbations, its modulus of continuity is $O(δ^ζ\log δ)$. This article has been accepted for publication in the Discrete and Continuous Dynamical Systems journal.

Quantitative statistical stability for the equilibrium states of piecewise partially hyperbolic maps

TL;DR

The paper addresses how equilibrium states for a class of skew-product maps , with a non-uniformly expanding base and a contracting fiber, respond to deterministic perturbations. It develops an anisotropic space for disintegrated measures and harnesses Hölder regularity of disintegration to obtain a spectral-gap framework for the transfer operator , enabling quantitative statistical stability. The authors prove existence and uniqueness of the invariant measure in and establish exponential convergence to equilibrium, then formulate and apply a perturbation theory via an -family of operators to obtain a modulus of continuity for the perturbed invariant measures. This yields precise, rate-based stability results for a broad class of deterministic perturbations, contributing to the understanding of uncertainty propagation in complex non-invertible partially hyperbolic systems.

Abstract

We consider a class of endomorphisms that contains a set of piecewise partially hyperbolic dynamics semi-conjugated to non-uniformly expanding maps. Our goal is to study a class of endomorphisms that preserve a foliation that is almost everywhere uniformly contracted, with possible discontinuity sets parallel to the contracting direction. We apply the spectral gap property and the -Hölder regularity of the disintegration of its equilibrium states to prove a quantitative statistical stability statement. More precisely, under deterministic perturbations of the system of size , we show that the -invariant measure varies continuously with respect to a suitable anisotropic norm. Moreover, we prove that for certain interesting classes of perturbations, its modulus of continuity is . This article has been accepted for publication in the Discrete and Continuous Dynamical Systems journal.

Paper Structure

This paper contains 13 sections, 25 theorems, 124 equations, 1 figure.

Table of Contents

  1. Examples
  2. Preliminary Results
  3. Weak and Strong Spaces
  4. $L^{\infty}$-like spaces.
  5. The $\mathcal{L}^{\infty}$ and $S^\infty$ spaces
  6. The transfer operator associated to $F$
  7. Basic properties of the norms and convergence to equilibrium
  8. Convergence to equilibrium
  9. Hölder-Measures
  10. Properties of Admissible $R({ \if@compatibility \mathchar"010E {} \mathchar"010E })$-Perturbations
  11. Perturbation of Operators
  12. Proof of Theorem \ref{['belongss']}
  13. Proof of Theorem \ref{['htyttigu']} and Corollary \ref{['htyttigui']}

Key Result

Theorem A

The system $F$ has a unique invariant probability, ${ \if@compatibility \mathchar"0116 {} \mathchar"0116 }_0 \in S^{\infty }$. If $F$ is continuous, then ${ \if@compatibility \mathchar"0116 {} \mathchar"0116 }_0$ is an equilibrium state.

Figures (1)

  • Figure 1: The graph of the perturbed map $g$.

Theorems & Definitions (57)

  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem A
  • Theorem B: Quantitative stability for deterministic perturbations
  • Corollary 1: Quantitative stability for deterministic perturbations with a linear $R({ \if@compatibility \mathchar"010E {} \mathchar"010E })$
  • Example 1.1
  • Example 1.2
  • Example 1.3
  • Example 1.4
  • ...and 47 more