# On the Orbits of not Expansive Mappings in Metric Spaces

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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.

Author: Sergio Venturini

Source: https://archive.org/