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Random Dynamical Systems on the circle without a finite orbit

Dominique Malicet, Graccyela Salcedo

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.

Random Dynamical Systems on the circle without a finite orbit

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.
Paper Structure (7 sections, 22 theorems, 70 equations, 1 figure)

This paper contains 7 sections, 22 theorems, 70 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Basic notations
  3. Invariant minimal sets
  4. Weight maps
  5. The neighbors
  6. Proximal action
  7. The inverse RDS

Key Result

Theorem 1

Let $\mathcal{F}\subset Hom(S^1)$ such that the generated semigroup $G_{\mathcal{F}}$ has no finite orbit. Let $K_1$,...,$K_d$ be the invariant minimal sets. Let us assume that $d\geq 2$. Then there exist subsets $L_1,\ldots, L_d$ of $\mathbb{S}^1$ such that Moreover, if $\mathcal{F}\subset Hom^+(S^1)$, the sets $L_i$ has the same number of connected components.

Figures (1)

  • Figure 1: Dynamics on the circle of the functions $f_1$, $f_2$ and $f_3$ in Example \ref{['ex1']} respectively.

Theorems & Definitions (48)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 1
  • Proposition 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • ...and 38 more