Expansivity and Cone-fields in Metric Spaces

Abstract

Due to the results of Lewowicz and Tolosa expansivity can be characterized with the aid of Lyapunov function. In this paper we study a similar problem for uniform expansivity and show that it can be described using generalized cone-fields on metric spaces. We say that a function $$f:X\rightarrow X$$ is uniformly expansive on a set $$\varLambda \subset X$$ if there exist $$\varepsilon >0$$ and $$\alpha \in (0,1)$$ such that for any two orbits $$\hbox {x}:\{-N,\ldots ,N\} \rightarrow \varLambda $$ , $$\hbox {v}:\{-N,\ldots ,N\} \rightarrow X$$ of $$f$$ we have $$\begin{aligned} \sup _{-N\le n\le N}d(\hbox {x}_n,\hbox {v}_n) \le \varepsilon \implies d(\hbox {x}_0,\hbox {v}_0) \le \alpha \sup _{-N\le n\le N}d(\hbox {x}_n,\hbox {v}_n). \end{aligned}$$ It occurs that a function is uniformly expansive iff there exists a generalized cone-field on $$X$$ such that $$f$$ is cone-hyperbolic.