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Modulus of continuity of invariant densities and entropies for piecewise expanding maps

José F. Alves, Odaudu Etubi

TL;DR

The paper investigates how invariant densities and metric entropies of physical (SRB/ACIP) measures for piecewise expanding maps depend on perturbations. It combines a general BV-transfer-operator framework with Galatolo–Lucena stability to obtain quantitative modulus-of-continuity results, showing that both densities and entropies vary with modulus $|t-s|\,|\log|t-s||$ under suitable perturbations. The authors apply the abstract results to a family of two-dimensional tent maps, verifying the required assumptions and establishing explicit bounds, thereby demonstrating Hölder-type continuity in a concrete, high-dimensional setting. The work advances understanding of statistical-stability for non-uniformly hyperbolic, piecewise smooth systems and provides sharp, computable bounds for the robustness of ACIPs and their entropies under perturbations.

Abstract

Using a perturbation result established by Galatolo and Lucena, we obtain quantitative estimates on the continuity of the invariant densities and entropies of the physical measures for some families of piecewise expanding maps. We apply these results to a family of two-dimensional tent maps.

Modulus of continuity of invariant densities and entropies for piecewise expanding maps

TL;DR

The paper investigates how invariant densities and metric entropies of physical (SRB/ACIP) measures for piecewise expanding maps depend on perturbations. It combines a general BV-transfer-operator framework with Galatolo–Lucena stability to obtain quantitative modulus-of-continuity results, showing that both densities and entropies vary with modulus under suitable perturbations. The authors apply the abstract results to a family of two-dimensional tent maps, verifying the required assumptions and establishing explicit bounds, thereby demonstrating Hölder-type continuity in a concrete, high-dimensional setting. The work advances understanding of statistical-stability for non-uniformly hyperbolic, piecewise smooth systems and provides sharp, computable bounds for the robustness of ACIPs and their entropies under perturbations.

Abstract

Using a perturbation result established by Galatolo and Lucena, we obtain quantitative estimates on the continuity of the invariant densities and entropies of the physical measures for some families of piecewise expanding maps. We apply these results to a family of two-dimensional tent maps.
Paper Structure (8 sections, 6 theorems, 67 equations, 1 figure)

This paper contains 8 sections, 6 theorems, 67 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Modulus of continuity
  3. Two-dimensional tent maps
  4. Functions of bounded variation
  5. Transfer operators
  6. Modulus of continuity for the densities
  7. Modulus of continuity for the entropies
  8. Application to tent maps

Key Result

Theorem A

Let $(\phi_t)_{t\in I}$ be a family of maps for which (U) holds and each $\phi_t$ has a unique ergodic absolutely continuous invariant probability measure $\mu_t$. Assume that there exists a function $\mathcal{E}:A\to\mathbb [0,1)$ such that, for all $s,t\in I$, Then, there exist $C>0$ and $0<\eta<1$ such that, for all $s,t\in I$,

Figures (1)

  • Figure 1: The tent maps

Theorems & Definitions (10)

  • Theorem A
  • Theorem B
  • Theorem C
  • Lemma 2.1
  • proof
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Remark 4.1