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Non uniform expansion and additive noise imply random horseshoe

Jeroen S. W. Lamb, Giuseppe Tenaglia, Dmitry Turaev

TL;DR

This work extends the random-dynamical-systems theory of Smale horseshoes to non-uniformly expanding maps with diffusive additive noise by proving the density of random horseshoes in the support of an ergodic stationary measure when all Lyapunov exponents are positive. The authors combine large-deviation estimates for Lyapunov-type averages with annealed control of large-ball dynamics and a refined Young-time framework to build a random Young tower, from which a κ-horseshoe is extracted. Key contributions include an annealed large-deviation scheme via a perturbed transfer operator, a density result for random horseshoes under mild regularity assumptions on the critical set, and a constructive horseshoe scheme relying on independent noise and stopping-time arguments. The findings generalize previous random-horseshoe results beyond predominantly expanding or uniformly hyperbolic settings, with implications for chaotic structure in stochastic dynamics. All mathematical notation is presented with explicit delimiters to ensure precise interpretation.

Abstract

We propose a notion of random horseshoe and prove density of random horseshoes for non uniformly expanding random dynamical systems with additive noise

Non uniform expansion and additive noise imply random horseshoe

TL;DR

This work extends the random-dynamical-systems theory of Smale horseshoes to non-uniformly expanding maps with diffusive additive noise by proving the density of random horseshoes in the support of an ergodic stationary measure when all Lyapunov exponents are positive. The authors combine large-deviation estimates for Lyapunov-type averages with annealed control of large-ball dynamics and a refined Young-time framework to build a random Young tower, from which a κ-horseshoe is extracted. Key contributions include an annealed large-deviation scheme via a perturbed transfer operator, a density result for random horseshoes under mild regularity assumptions on the critical set, and a constructive horseshoe scheme relying on independent noise and stopping-time arguments. The findings generalize previous random-horseshoe results beyond predominantly expanding or uniformly hyperbolic settings, with implications for chaotic structure in stochastic dynamics. All mathematical notation is presented with explicit delimiters to ensure precise interpretation.

Abstract

We propose a notion of random horseshoe and prove density of random horseshoes for non uniformly expanding random dynamical systems with additive noise
Paper Structure (9 sections, 14 theorems, 120 equations)

This paper contains 9 sections, 14 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. Statement of the result
  3. Hypotheses and main result
  4. Large Deviation estimates
  5. Annealed properties of orbits of large balls
  6. Existence of Young Times
  7. Existence of the horseshoe
  8. Preliminary work
  9. Horseshoe construction

Key Result

Theorem 2.2

Let $f^n_{\omega}$ the random composition in iterazioni satisfying Hypotheses (A)-(B). Then, there exists $\kappa>0$ such that the set of $\kappa$-horseshoes is dense, which means that for every open set $A$ there are balls $I_1,I_2 \Subset A$ such that $(I_1,I_2)$ is a $\kappa$-horseshoe.

Theorems & Definitions (28)

  • Definition 2.1
  • Theorem 2.2
  • Proposition 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Proposition 3.4
  • proof
  • Proposition 4.1
  • ...and 18 more