On the Orbits of not Expansive Mappings in Metric Spaces

Abstract

Let $\MSpace$ be a locally compact metric space and let $\pMap:\MSpace\to\MSpace$ be a not expansive map. We prove that for each $\ppa_0\in\MSpace$ the sequence $\ppa_0,\pMap(\ppa_0),\pMap^2(\ppa_0),\ldots$ is either relatively compact in $\MSpace$ or compactly divergent in $\MSpace$. As applications we study the structure of the functions which are limits of the iterates of the map $\pMap$ and we prove the analyticity of the set of $\pMap$-recurrent points when $\pMap:\MSpace\to\MSpace$ is a holomorphic and $\MSpace$ is a complex hyperbolic spaces in the sense of Kobayashi.