A family of rotation numbers for discrete random dynamics on the circle

TL;DR

This paper investigates rotation numbers in discrete random circle dynamics, revealing differences from deterministic cases, and provides formulas and sampling theorems to connect various rotation number concepts.

Contribution

It introduces a comprehensive analysis of rotation numbers for random systems, including their dependence on lifts and a sampling theorem linking discrete and continuous dynamics.

Findings

Topological rotation numbers depend on the choice of lifts in random systems.

Winding orbit rotation numbers differ from topological rotation numbers.

Sampling theorem connects discrete rotation numbers to continuous stochastic flows.

Abstract

We revisit the problem of well-defining rotation numbers for discrete random dynamical systems on the circle. We show that, contrasting with deterministic systems, the topological (i.e. based on Poincar\'{e} lifts) approach does depend on the choice of lifts (e.g. continuously for nonatomic randomness). Furthermore, the winding orbit rotation number does not agree with the topological rotation number. Existence and conversion formulae between these distinct numbers are presented. Finally, we prove a sampling in time theorem which recover the rotation number of continuous Stratonovich stochastic dynamical systems on $S^1$ out of its time discretisation of the flow.