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

通过 Lipschitz 自由空间的线性化特性，两个指向度量空间之间的任何 Lipschitz 映射\(f : M \rightarrow N\)可以唯一地扩展到有界线性算子\({\widehat{f}} : {\mathcal {F}}(M) \rightarrow {\mathcal {F}}(N)\)在它们对应的 Lipschitz-free 空间之间。在这篇笔记中，我们探讨了 Lipschitz 自映射的动力学\(f : M \rightarrow M\)与其扩展的线性动力学\({\widehat{f}} : {\mathcal {F}} (M) \rightarrow {\mathcal {F}}(M)\)。这不仅使我们能够将拓扑动力系统与线性动力系统联系起来，而且还提供了一类新的作用于 Lipschitz 自由空间的超循环算子。