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Periodicity and Rotation Number for Random Circle Homeomorphisms

Zixu Li, Simon Lloyd

Abstract

We study discrete-time random dynamical systems where each fibre map is an orientation-preserving homeomorphism of the circle. We prove that the existence of a random periodic cycle with period at least two implies that the random rotation number is rational almost surely. Moreover, in clear contrast with the deterministic setting, we demonstrate that a common fixed point for the fibre maps does not imply that the random rotation number is an integer. Conversely, we show that if the mean random rotation number is an integer, then the fibre maps have a fixed point with positive probability.

Periodicity and Rotation Number for Random Circle Homeomorphisms

Abstract

We study discrete-time random dynamical systems where each fibre map is an orientation-preserving homeomorphism of the circle. We prove that the existence of a random periodic cycle with period at least two implies that the random rotation number is rational almost surely. Moreover, in clear contrast with the deterministic setting, we demonstrate that a common fixed point for the fibre maps does not imply that the random rotation number is an integer. Conversely, we show that if the mean random rotation number is an integer, then the fibre maps have a fixed point with positive probability.
Paper Structure (4 sections, 5 theorems, 51 equations, 3 figures)

This paper contains 4 sections, 5 theorems, 51 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Random circle homeomorphisms
  3. Random periodic cycles
  4. Integer-valued mean random rotation number

Key Result

Lemma 1

Let $F\in \tilde{\mathcal{H}}(\Omega)$ be the standard random lift of a random order-preserving circle map $f\in \mathcal{H}(\Omega)$. The random lift $F$ satisfies for all $x\in \mathbb{R}$ and all $\omega\in \Omega$,

Figures (3)

  • Figure 1: Graphs of the circle homeomorphisms $g_0$ (solid line) and $g_1$ (dashed line).
  • Figure 2: Graphs of the fibre map $f_\omega$ when $\omega_0=0$ (solid line) and when $\omega_0=1$ (dashed line).
  • Figure 3: Graphs of the fibre map $f_\omega$ when $\omega=1$ (solid line) and when $\omega=2$ (dashed line).

Theorems & Definitions (17)

  • Lemma 1
  • proof
  • Lemma 2
  • Proposition 3
  • proof
  • Definition 4
  • Example 5
  • Proposition 6
  • proof
  • proof
  • ...and 7 more