On the Orbits of not Expansive Mappings in Metric Spaces

TL;DR

This paper investigates the behavior of orbits under not expansive maps in locally compact metric spaces, showing they are either relatively compact or diverge, and explores the structure of limit functions and recurrence in complex hyperbolic spaces.

Contribution

It establishes a dichotomy for orbits of not expansive maps and analyzes the structure of limit functions and recurrence sets in complex hyperbolic spaces.

Findings

Orbits are either relatively compact or diverge.

Limit functions of iterates have a specific structure.

Recurrence sets are analytic in complex hyperbolic 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.