Kitai's Criterion for Composition Operators

TL;DR

This paper develops a unified framework to analyze the dynamics of composition operators on measurable function spaces, clarifying conditions for hypercyclicity, mixing, and Kitai's Criterion, with new characterizations and examples.

Contribution

It introduces a general framework for composition operator dynamics, reexamines key properties, and provides new characterizations and examples related to Kitai's Criterion.

Findings

Hypercyclicity and weak mixing coincide in this context.

Recurrent composition operators match hypercyclic ones in dissipative systems.

A new characterization for invertible composition operators satisfying Kitai's Criterion.

Abstract

We present a general and natural framework to study the dynamics of composition operators on spaces of measurable functions, in which we then reconsider the characterizations for hypercyclic and mixing composition operators obtained by Bayart, Darji and Pires. We show that the notions of hypercyclicity and weak mixing coincide in this context and, if the system is dissipative, the recurrent composition operators agree with the hypercyclic ones. We also give a characterization for invertible composition operators satisfying Kitai's Criterion, and we construct an example of a mixing composition operator not satisfying Kitai's Criterion. For invertible dissipative systems with bounded distortion we show that composition operators satisfying Kitai's Criterion coincide with the mixing operators.