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A version of Oseledets for proximal random dynamical System on the circle

Jamerson Bezerra, Graccyela Salcedo

TL;DR

This work extends Oseledets-type theory to proximal random dynamical systems on the circle generated by circle homeomorphisms, proving the existence of random forward and backward directions $\\pi$ and $\\theta$ that govern asymptotic behavior. In the differentiable case, extremal Lyapunov exponents $\\Lambda(\\nu)$ and $\\lambda(\\nu)$ are characterized and linked to the stationary measures via $\\eta$ and $\\eta^{-}$, yielding precise growth rates for derivatives at the random directions. The authors establish exact dimensionality of the stationary measure, showing $\\operatorname{dim}(\\eta) = -\frac{h_F(\\eta,\\nu)}{\\lambda(\\nu)}$, and provide broad implications for synchronization and entropy in proximal circle RDSs. The results generalize nonlinear Oseledets-type behavior beyond linear cocycles, accommodate non-orientation-preserving maps, and include explicit examples of nonlinear proximal dynamics that resist linear conjugacy, highlighting the novelty and applicability of the framework.

Abstract

We study proximal random dynamical systems of homeomorphisms of the circle without a common fixed point. We prove the existence of two random points that govern the behavior of the forward and backward orbits of the system. Assuming the differentiability of the maps, we characterize these random points in terms of the extremal Lyapunov exponents of the random dynamical system. As an application, we prove the exactness of the stationary measure in this setting.

A version of Oseledets for proximal random dynamical System on the circle

TL;DR

This work extends Oseledets-type theory to proximal random dynamical systems on the circle generated by circle homeomorphisms, proving the existence of random forward and backward directions and that govern asymptotic behavior. In the differentiable case, extremal Lyapunov exponents and are characterized and linked to the stationary measures via and , yielding precise growth rates for derivatives at the random directions. The authors establish exact dimensionality of the stationary measure, showing , and provide broad implications for synchronization and entropy in proximal circle RDSs. The results generalize nonlinear Oseledets-type behavior beyond linear cocycles, accommodate non-orientation-preserving maps, and include explicit examples of nonlinear proximal dynamics that resist linear conjugacy, highlighting the novelty and applicability of the framework.

Abstract

We study proximal random dynamical systems of homeomorphisms of the circle without a common fixed point. We prove the existence of two random points that govern the behavior of the forward and backward orbits of the system. Assuming the differentiability of the maps, we characterize these random points in terms of the extremal Lyapunov exponents of the random dynamical system. As an application, we prove the exactness of the stationary measure in this setting.

Paper Structure

This paper contains 11 sections, 22 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Statement of the results
  3. Properties of proximal systems
  4. Random repelling point
  5. Random attracting point
  6. Proofs of Theorem \ref{['teo:01']} and Corollary \ref{['cor01:teo01']}
  7. Differentiable case
  8. Extremal Lyapunov exponents
  9. Proof of Theorem \ref{['teo:02']} and Corollary \ref{['cor:teo:02']}
  10. Exact Dimensionality
  11. examples

Key Result

Theorem 1

Let $\nu$ be a probability measure on $\operatorname{Hom}(\mathbb{S}^1)$. Assume that $\Gamma_{\nu}$ is proximal and does not fix any point in $\mathbb{S}^1$. Then there exist measurable functions $\pi,\theta: X_\nu^\mathbb{N}\to\mathbb{S}^1$ such that

Theorems & Definitions (47)

  • Theorem 1
  • Corollary 2
  • Theorem 3
  • Corollary 4
  • Theorem 5
  • Proposition 6
  • proof
  • Proposition 7
  • proof
  • Remark 8
  • ...and 37 more