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On the dynamics of Lipschitz operators. (English) Zbl 1495.47022

Every Lipschitz map \(f: M \rightarrow N\) between two pointed metric spaces may be extended uniquely to a continuous linear operator \(\hat{f}: \mathcal{F}(M) \rightarrow \mathcal{F}(N)\) between their corresponding Lipschitz-free spaces. The authors investigate the connections between the topological dynamics of Lipschitz self-maps \(f: M \rightarrow N\) and the linear dynamics of their extensions \(\hat{f}: \mathcal{F}(M) \rightarrow \mathcal{F}(N)\). Several properties like chaos, (weakly) mixing condition and hypercyclicity are considered. A new class of hypercyclic operators acting on Lipschitz-free spaces is also presented.

MSC:

47A16 Cyclic vectors, hypercyclic and chaotic operators
46B20 Geometry and structure of normed linear spaces
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References:

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