On the dynamics of Lipschitz operators

Abstract

By the linearization property of Lipschitz-free spaces, any Lipschitz map $f:M\to N$ between two pointed metric spaces may be extended uniquely to a bounded linear operator $\hat{f}:\mathcal{F}(M)\to \mathcal{F}(N)$ between their corresponding Lipschitz-free spaces. In this note, we explore the connections between the dynamics of Lipschitz self-maps $f:M\to M$ and the linear dynamics of their extensions $\hat{f}:\mathcal{F}(M)\to \mathcal{F}(M)$. This not only allows us to relate topological dynamical systems to linear dynamical systems but also provide a new class of hypercyclic operators acting on Lipschitz-free spaces.