Random Dynamical Systems on the circle without a finite orbit

TL;DR

This paper investigates the topological and measure-theoretic properties of random dynamical systems on the circle that lack finite orbits, focusing on invariant sets, accumulation points, and relationships between minimal sets of the system and its inverse.

Contribution

It provides a characterization of invariant sets, analyzes accumulation points of the transfer operator, and explores properties of ergodic stationary measures for such systems.

Findings

Invariant sets are finite unions of intervals.

Accumulation points of the transfer operator are described.

Properties of weight functions for ergodic stationary measures are identified.

Abstract

In this paper, we study Random Dynamical Systems (RDSs) of homeomorphisms on the circle without a finite orbit. We characterize the topological dynamics of the associated semigroup by identifying the existence of invariant sets which are finite unions of intervals. We describe the accumulation points of the average orbit of the transfer operator. For each ergodic stationary measure, we demonstrate interesting properties of its weight function on the circle. Relationships between the minimal sets of an RDS and its inverse RDS are also established.