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Unique ergodicity for noninvertible function systems on an interval

Sander C. Hille, Hanna Oppelmayer, Tomasz Szarek

Abstract

We study random dynamical systems of certain continuous functions on the unit interval. We use bounded variation to provide sufficient conditions for unique ergodicity of these systems. Several classes of examples are provided.

Unique ergodicity for noninvertible function systems on an interval

Abstract

We study random dynamical systems of certain continuous functions on the unit interval. We use bounded variation to provide sufficient conditions for unique ergodicity of these systems. Several classes of examples are provided.

Paper Structure

This paper contains 9 sections, 16 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some measure-theoretic auxiliary results
  4. Proofs of Theorem \ref{['thm:main result']} and Corollary \ref{['clry:main application']} & \ref{['cor1_16.09.68']}
  5. Various sufficient conditions and examples
  6. Monotonic maps
  7. $\mu$--injective maps
  8. Systems constructed by tableaux
  9. Compensation of unbounded variation.

Key Result

Theorem 1.1

Let $(\Gamma, \mu)$ be a stochastic dynamical system on $C([0, 1])$ such that property BV is satisfied. Then the following statements hold:

Theorems & Definitions (32)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Remark 2.1
  • Remark 2.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 22 more