Preprint D84/2011
A generalization of expansivity

C. A. Morales

**Keywords: **
Positively n-expansive; n-expansive; Metric Space

We study dynamical systems for which at most $n$ orbits can accompany a given arbitrary orbit. For simplicity we call them {\em $n$-expansive} (or {\em positively $n$-expansive} if positive orbits are considered instead). We prove that these systems can satisfy properties of expansive systems or not. For instance, unlike positively expansive maps \cite{ck}, positively $n$-expansive homeomorphisms may exist on certain infinite compact metric spaces. We also prove that a map (resp. bijective map) is positively $n$-expansive (resp. $n$-expansive) if and only if it is so outside finitely many points. Finally, we prove that a homeomorphism on a compact metric space is $n$-expansive if and only if it is so outside finitely many orbits. These last results extends previous ones about expansive systems \cite{br},\cite{u'},\cite{w}.