Mean ergodic composition operators on spaces of smooth functions and distributions

TL;DR

This paper studies the mean ergodic behavior of weighted composition operators on smooth functions and distributions, revealing that such operators with real analytic diffeomorphic symbols are mean ergodic only if they are periodic with period 2.

Contribution

It provides a characterization of mean ergodicity for composition operators on Montel spaces, linking it to periodicity and growth properties of orbits, with new insights into operators on distributions.

Findings

Composition operators with real analytic symbols are mean ergodic only if periodic with period 2.

Characterization of mean ergodicity via Cesàro boundedness and orbit growth.

New results on the behavior of operators on spaces of distributions.

Abstract

We investigate (uniform) mean ergodicity of weighted composition operators on the space of smooth functions and the space of distributions, both over an open subset of the real line. Among other things, we prove that a composition operator with a real analytic diffeomorphic symbol is mean ergodic on the space of distributions if and only if it is periodic with period 2. Our results are based on a characterization of mean ergodicity in terms of Ces\`aro boundedness and a growth property of the orbits for operators on Montel spaces which is of independent interest.