Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Note on the Uniform Ergodicity of Dynamical Systems

Julian Hölz

Abstract

We study the uniform ergodicity property for non-invertible topological and measure-preserving dynamical systems. It is shown that for topological dynamical systems uniform ergodicity is equivalent to eventually periodicity and that for measure preserving systems it is equivalent to periodicity. To obtain our results, we prove a result on the long-term behavior of lattice homomorphisms that have $1$ isolated in its spectrum.

A Note on the Uniform Ergodicity of Dynamical Systems

Abstract

We study the uniform ergodicity property for non-invertible topological and measure-preserving dynamical systems. It is shown that for topological dynamical systems uniform ergodicity is equivalent to eventually periodicity and that for measure preserving systems it is equivalent to periodicity. To obtain our results, we prove a result on the long-term behavior of lattice homomorphisms that have isolated in its spectrum.
Paper Structure (3 sections, 11 theorems, 31 equations)

This paper contains 3 sections, 11 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Uniform Almost Periodicity and Uniform Ergodicity
  3. Application to Composition Operators

Key Result

Theorem 2.1

Let $E$ be a complex Banach lattice and $T: E \to E$ be linear and a lattice homomorphism that has spectral radius $r(T) = 1$. If $1$ is an isolated value in the spectrum $\sigma(T)$, then $T$ is power-bounded and uniformly ergodic. Moreover, there exist a closed lattice ideal $I_{\mathrm{stab}}$ an and Moreover, the peripheral spectrum $\sigma(T) \cap \mathbb{T}$ consists of a finite union of ro

Theorems & Definitions (30)

  • Theorem 2.1
  • Lemma 2.2
  • proof
  • proof : Proof of Theorem \ref{['theorem:uniform-almost-periodicity-isolated-spectral-value']}
  • Remark 2.3
  • Example 2.4
  • Corollary 2.5
  • proof
  • Corollary 2.6
  • proof : Proof of Corollary \ref{['corollary:almost-uniform-periodicity']}
  • ...and 20 more