Expansivity and strong structural stability for composition operators on $L^p$ spaces

TL;DR

This paper characterizes when composition operators on L^p spaces are expansive or strongly structurally stable, establishing conditions and relationships between these properties, especially in dissipative cases.

Contribution

It provides necessary and sufficient conditions for expansivity and links shadowing property with strong structural stability in dissipative composition operators.

Findings

Expansivity characterized by specific conditions.

Shadowing property implies strong structural stability in dissipative cases.

Expansivity and strong structural stability are equivalent under positive expansivity.

Abstract

In this note we investigate the two notions of expansivity and strong structural stability for composition operators on $L^p$ spaces, $1 \leq p < \infty$. Necessary and sufficient conditions for such operators to be expansive are provided, both in the general and the dissipative case. We also show that, in the dissipative setting, the shadowing property implies the strong structural stability and we prove that these two notions are equivalent under the extra hypothesis of positive expansivity.