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Limit Theorems and Quantitative Statistical Stability for the Equilibrium States of Piecewise Partially Hyperbolic Maps

Rafael A. Bilbao, Rafael Lucena

TL;DR

The paper develops a flexible transfer-operator framework for piecewise, non-continuous, non-invertible skew-product maps $F(x,y)=(f(x),G(x,y))$ with non-uniform base expansion and fiber contraction. It establishes a spectral-gap structure on a hierarchy of spaces, yielding exponential decay of correlations and a Central Limit Theorem for Hölder observables, while also providing a quantitative theory of statistical stability under admissible perturbations. By avoiding compact embeddings, the approach handles singularities and discontinuities that thwart classical Lasota–Yorke techniques, and it extends to a broad class of potentials via homologous reductions. The results are illustrated with partially hyperbolic horseshoes, non-invertible maps semi-conjugate to Manneville–Pomeau dynamics, and fat solenoidal attractors, highlighting both the reach and the robustness of the method. Overall, the work delivers explicit modulus-of-continuity bounds for equilibrium states under perturbations and establishes a comprehensive probabilistic description for a wide family of discontinuous, non-invertible dynamical systems.

Abstract

This paper establishes limit theorems and quantitative statistical stability for a class of piecewise partially hyperbolic maps that are neither necessarily continuous nor locally invertible. By employing a flexible functional-analytic framework that bypasses the classical requirement of compact embeddings between Banach spaces, we obtain explicit rates of convergence for the variation of equilibrium states under perturbations. Furthermore, we prove the exponential decay of correlations and the Central Limit Theorem for Hölder observables. A key feature of our approach is its applicability to systems where traditional spectral gap techniques fail due to the presence of singularities and the lack of invertibility. We provide several examples illustrating the scope of our results, including partially hyperbolic attractors over horseshoes, non-invertible dynamics semi-conjugated to Manneville--Pomeau maps, and fat solenoidal attractors.

Limit Theorems and Quantitative Statistical Stability for the Equilibrium States of Piecewise Partially Hyperbolic Maps

TL;DR

The paper develops a flexible transfer-operator framework for piecewise, non-continuous, non-invertible skew-product maps with non-uniform base expansion and fiber contraction. It establishes a spectral-gap structure on a hierarchy of spaces, yielding exponential decay of correlations and a Central Limit Theorem for Hölder observables, while also providing a quantitative theory of statistical stability under admissible perturbations. By avoiding compact embeddings, the approach handles singularities and discontinuities that thwart classical Lasota–Yorke techniques, and it extends to a broad class of potentials via homologous reductions. The results are illustrated with partially hyperbolic horseshoes, non-invertible maps semi-conjugate to Manneville–Pomeau dynamics, and fat solenoidal attractors, highlighting both the reach and the robustness of the method. Overall, the work delivers explicit modulus-of-continuity bounds for equilibrium states under perturbations and establishes a comprehensive probabilistic description for a wide family of discontinuous, non-invertible dynamical systems.

Abstract

This paper establishes limit theorems and quantitative statistical stability for a class of piecewise partially hyperbolic maps that are neither necessarily continuous nor locally invertible. By employing a flexible functional-analytic framework that bypasses the classical requirement of compact embeddings between Banach spaces, we obtain explicit rates of convergence for the variation of equilibrium states under perturbations. Furthermore, we prove the exponential decay of correlations and the Central Limit Theorem for Hölder observables. A key feature of our approach is its applicability to systems where traditional spectral gap techniques fail due to the presence of singularities and the lack of invertibility. We provide several examples illustrating the scope of our results, including partially hyperbolic attractors over horseshoes, non-invertible dynamics semi-conjugated to Manneville--Pomeau maps, and fat solenoidal attractors.
Paper Structure (26 sections, 41 theorems, 160 equations, 1 figure)

This paper contains 26 sections, 41 theorems, 160 equations, 1 figure.

Table of Contents

  1. Settings and Examples
  2. Hypothesis on the basis map $f$.
  3. Spectral gap for $\mathcal{L}_{ \if@compatibility \mathchar"0127 {} \mathchar"0127 }$
  4. Hypothesis on the fiber map $G$.
  5. Examples.
  6. Preliminary Concepts
  7. The W-K like norm
  8. Rokhlin's Disintegration Theorem.
  9. The Space $\mathcal{AB}_m$.
  10. Hölder-Measures.
  11. The $\mathcal{L}_m^{\infty}$ and $S^\infty$ spaces.
  12. The actions on $\mathcal{H} ^{ \if@compatibility \mathchar"0110 {} \mathchar"0110 }_m$, $\mathcal{L}^\infty _m$ and $S^\infty$.
  13. The operator
  14. On the actions of $\mathop{\rm \overline{F}}\nolimits_{\Phi}$ on $\mathcal{H}^{ \if@compatibility \mathchar"0110 {} \mathchar"0110 } _m$, $\mathcal{L}^\infty$ and $S^\infty$.
  15. On the actions of $\mathop{\rm \overline{F}}\nolimits_{\Phi}$ on $\mathcal{L}^\infty$ and $S^\infty$.
  16. ...and 11 more sections

Key Result

Theorem 1.1

There exist constants $B_1>0$, $C_1>0$ and $0<{ \if@compatibility \mathchar"010C {} \mathchar"010C }_1<1$ such that for all $u \in B_s$, and all $n \geq 1$, it holds

Figures (1)

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

Theorems & Definitions (92)

  • Definition 1
  • Theorem 1.1
  • Theorem 1.2
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Proposition 1.3
  • Remark 5
  • Example 1.1
  • ...and 82 more