On the dynamics of Lipschitz operators

TL;DR

This paper investigates the relationship between the dynamics of Lipschitz maps on metric spaces and their linear extensions on Lipschitz-free spaces, revealing new hypercyclic operators and linking topological and linear dynamics.

Contribution

It establishes a connection between the dynamics of Lipschitz maps and their linear extensions, introducing a new class of hypercyclic operators on Lipschitz-free spaces.

Findings

Identifies a correspondence between topological and linear dynamical systems.

Introduces new hypercyclic operators on Lipschitz-free spaces.

Provides insights into 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 $\widehat{f} : \mathcal F(M) \to \mathcal F(N)$ 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 $\widehat{f} : \mathcal F(M) \to \mathcal F(M)$. 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.