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Lift-Free Approaches to Random Rotation Number and Numerical Approximation

Zixu Li, Simon Lloyd

Abstract

We study the random rotation number for random circle homeomorphisms. We introduce two new definitions of the random rotation number that can be stated without reference to any choice of lift of the dynamics to the real line, and prove that they are equivalent to the standard random rotation number. We then prove that the mean random rotation number may be approximated within an error of $1/n$ when using $n$ iterations of the dynamics. Finally, we develop numerical algorithms for approximation of the random rotation number which we test with several examples.

Lift-Free Approaches to Random Rotation Number and Numerical Approximation

Abstract

We study the random rotation number for random circle homeomorphisms. We introduce two new definitions of the random rotation number that can be stated without reference to any choice of lift of the dynamics to the real line, and prove that they are equivalent to the standard random rotation number. We then prove that the mean random rotation number may be approximated within an error of when using iterations of the dynamics. Finally, we develop numerical algorithms for approximation of the random rotation number which we test with several examples.

Paper Structure

This paper contains 17 sections, 5 theorems, 73 equations, 6 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Background and notation
  3. Orientation-preserving circle homeomorphisms
  4. Random circle homeomorphisms
  5. Skew product formulation and acceleration
  6. Lift-free approaches to random rotation number
  7. Lift dependence of the random rotation number
  8. Binary coding
  9. Visit counting
  10. Error bounds for the mean random rotation number
  11. Numerical Algorithms
  12. Classical approach
  13. Binary coding approach
  14. Visit counting approach
  15. Comparison of methods
  16. ...and 2 more sections

Key Result

Theorem 2

Let $(\Omega, \mathcal{F}, \mathbb{P}, \sigma)$ be a measure-preserving dynamical system. Let $f\in \mathcal{H}(\Omega)$ and let $F\in\tilde{\mathcal{H}}$ be the standard lift of $f$. Then, for each $x_0\in [0,1)$ and $n\in\mathbb{N}$, and for $\mathbb{P}$-almost every $\omega\in\Omega$,

Figures (6)

  • Figure 1: Graph showing approximations $A_n$ of the random rotation number against $n$, and the region bounded by the curves $\pm1/n$ (light grey).
  • Figure 2: Graph showing approximations of the random rotation number against $n$ based on a single trajectory (top) and by averaging over 100 trajectories (bottom), along with the error bounds $1/4\pm1/n$ (light grey).
  • Figure 3: Graph of the interval exchange transformation $\sigma$ defined in (\ref{['eqn:iet']}).
  • Figure 4: Graphs of $\alpha(\omega)$ and $\beta(\omega)$ against $\omega\in[0,1)$ (left) and $F_\omega(x)$ against $x\in[0,1)$ for selected $\omega$ (right).
  • Figure 5: Graph showing approximations $R_F(n,100,0)$ of the random rotation number against $n$, along with the estimated error bounds $R_F(1000,100,0)\pm1/n$ (light grey).
  • ...and 1 more figures

Theorems & Definitions (15)

  • Example 1
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Example 4
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • ...and 5 more