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Stability and limit theorems in random dynamical systems

Davi Lima, Rafael Lucena

Abstract

The robust statistical description of dynamical systems under perturbations is a central problem in ergodic theory. In this paper, we investigate the statistical properties of skew-product maps driven by a subshift of finite type with contracting fiber maps, a setting that naturally encompasses Iterated Function Systems (IFS) and Random Dynamical Systems (RDS). Diverging from the classical perturbative frameworks that rely on the compact embedding of anisotropic Banach spaces, we employ a flexible operator approach based on the Lipschitz regularity of the invariant measure's disintegrations with respect to the Wasserstein metric. Our main results are threefold: first, we prove the quantitative statistical stability of the unique invariant measure under admissible deterministic perturbations, obtaining an explicit modulus of continuity of the form $O(R(δ) \log δ)$. Second, we establish the exponential decay of correlations on new pair of spaces of observables. Finally, leveraging this exponential decay and Gordin's method, we prove the Central Limit Theorem for the fluctuations of Birkhoff averages of Lipschitz observables.

Stability and limit theorems in random dynamical systems

Abstract

The robust statistical description of dynamical systems under perturbations is a central problem in ergodic theory. In this paper, we investigate the statistical properties of skew-product maps driven by a subshift of finite type with contracting fiber maps, a setting that naturally encompasses Iterated Function Systems (IFS) and Random Dynamical Systems (RDS). Diverging from the classical perturbative frameworks that rely on the compact embedding of anisotropic Banach spaces, we employ a flexible operator approach based on the Lipschitz regularity of the invariant measure's disintegrations with respect to the Wasserstein metric. Our main results are threefold: first, we prove the quantitative statistical stability of the unique invariant measure under admissible deterministic perturbations, obtaining an explicit modulus of continuity of the form . Second, we establish the exponential decay of correlations on new pair of spaces of observables. Finally, leveraging this exponential decay and Gordin's method, we prove the Central Limit Theorem for the fluctuations of Birkhoff averages of Lipschitz observables.
Paper Structure (18 sections, 31 theorems, 111 equations)

This paper contains 18 sections, 31 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. Foundational Framework and Notation
  3. Sub-shifts of Finite Type and Transfer Operators
  4. Dynamics of Contracting Fiber Maps
  5. Assumptions on the Fiber Map $G$
  6. Functional Framework and Measure Disintegration
  7. Foundations of Rokhlin's Disintegration
  8. The Space of Lipschitz Measures: $\mathcal{L}_{ \if@compatibility \mathchar"0112 {} \mathchar"0112 }$
  9. The spaces $\mathcal{L}^\infty$ and $S^\infty$
  10. The Transfer Operator and its Functional Properties
  11. Statistical Stability
  12. Admissible $R({ \if@compatibility \mathchar"010E {} \mathchar"010E })$-Perturbations
  13. Perturbation Theory for Operators
  14. Exponential Decay of Correlations and Central Limit Theorem
  15. Exponential Decay of Correlations over constant fiber functions
  16. ...and 3 more sections

Key Result

Proposition 2.1

One can find a constant $0 < { \if@compatibility \mathchar"011C {} \mathchar"011C }_2 < 1$ such that, for every ${ \if@compatibility \mathchar"0120 {} \mathchar"0120 } \in L^1_{m}(\Sigma_A^+)$ and ${ \if@compatibility \mathchar"0127 {} \mathchar"0127 } \in \mathcal{F}_{ \if@comp where $D({ \if@compatibility \mathchar"0120 {} \mathchar"0120 }, { \if@compatibility \math

Theorems & Definitions (64)

  • Proposition 2.1
  • Example 2.1
  • Theorem 3.1
  • Lemma 3.2
  • Definition 1
  • Lemma 3.3
  • Remark 1
  • Definition 2
  • Remark 2
  • Definition 3
  • ...and 54 more