On the Dynamics of Lipschitz Operators

By the linearization property of Lipschitz-free spaces, any Lipschitz map f : M -> N between two pointed metric spaces may be extended uniquely to a bounded linear operator (f) over cap : F(M) -> F(N) between their corresponding Lipschitz-free spaces. In this note, we explore the connections between the dynamics of Lipschitz self-maps f : M -> M and the linear dynamics of their extensions (f) over cap : F(M) -> 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.