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arXiv · 2011.10800

On the dynamics of Lipschitz operators

Arafat Abbar·Clément Coine·Colin Petitjean
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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.

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Arafat Abbar, Clément Coine, Colin Petitjean. 2020-11-21. On the dynamics of Lipschitz operators. https://arxiv.org/abs/2011.10800

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