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On the Dynamics of Lipschitz Operators

Authors
  • Abbar, Arafat1
  • Coine, Clément2
  • Petitjean, Colin1
  • 1 Univ Gustave Eiffel, Univ Paris Est Creteil, Marne-la-Vallée, 77447, France , Marne-la-Vallée (France)
  • 2 Normandie Univ, UNICAEN, Caen, 14000, France , Caen (France)
Type
Published Article
Journal
Integral Equations and Operator Theory
Publisher
Springer International Publishing
Publication Date
Jul 10, 2021
Volume
93
Issue
4
Identifiers
DOI: 10.1007/s00020-021-02662-4
Source
Springer Nature
Keywords
Disciplines
  • Article
License
Yellow

Abstract

By the linearization property of Lipschitz-free spaces, any Lipschitz map f:M→N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f : M \rightarrow N$$\end{document} between two pointed metric spaces may be extended uniquely to a bounded linear operator f^:F(M)→F(N)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\widehat{f}} : {\mathcal {F}}(M) \rightarrow {\mathcal {F}}(N)$$\end{document} between their corresponding Lipschitz-free spaces. In this note, we explore the connections between the dynamics of Lipschitz self-maps f:M→M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f : M \rightarrow M$$\end{document} and the linear dynamics of their extensions f^:F(M)→F(M)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\widehat{f}} : {\mathcal {F}}(M) \rightarrow {\mathcal {F}}(M)$$\end{document}. 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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