A generalization of expansivity (Q765050)

The aim of the present paper is to study dynamical systems for which at most \(n\) orbits can accompany a given arbitrary orbit. For simplicity these system are called \(n\)-expansive. The authors prove conditions under which these systems can satisfy properties of expansive systems. It is also proven that a map is positively \(n\)-expansive if and only if it is so outside of finitely many orbits. Finally, for homeomorphisms on compact metric spaces it is shown that \(n\)-expansivity is equivalent to being outside of finitely many orbits.