On the Dynamics of Weighted Composition Operators II

Abstract

We establish complete characterizations of various notions of expansivity for weighted composition operators on a very general class of locally convex spaces of continuous functions. This class includes several classical classes of continuous function spaces, such as the Banach spaces $C_0(X)$ of continuous scalar-valued functions vanishing at infinity on a Hausdorff locally compact space $X$, endowed with the sup norm, and the locally convex spaces $C(X)_c$ of continuous scalar-valued functions on a completely regular space $X$, endowed with the compact-open topology. We also obtain complete characterizations of various notions of expansivity for weighted composition operators on $L^p(μ)$ spaces, thereby complementing and extending previously known results in the unweighted case. Finally, we establish a conjugation between weighted and unweighted composition operators in the case of dissipative systems on $L^p(μ)$ spaces and apply it to the study of several dynamical properties.