Dynamics and spectra of composition operators on the Schwartz space

TL;DR

This paper investigates the dynamics and spectral properties of composition operators on the Schwartz space, revealing conditions for supercyclicity, power boundedness, and mean ergodicity, especially for polynomial symbols.

Contribution

It provides new characterizations of when composition operators on Schwartz space are power bounded, mean ergodic, or have specific spectral properties, focusing on polynomial symbols.

Findings

Composition operators are never supercyclic on Schwartz space.

Power boundedness occurs only in trivial cases for monotonic symbols.

Polynomial symbols of degree > 1 are mean ergodic iff they are power bounded and have even degree without fixed points.

Abstract

In this paper we study the dynamics of the composition operators defined in the Schwartz space $\mathcal{S}(\mathbb{R})$ of rapidly decreasing functions. We prove that such an operator is never supercyclic and, for monotonic symbols, it is power bounded only in trivial cases. For a polynomial symbol $\varphi$ of degree greater than one we show that the operator is mean ergodic if and only if it is power bounded and this is the case when $\varphi$ has even degree and lacks fixed points. We also discuss the spectrum of composition operators.