Nonstandard Expansiveness

Abstract

Let $(X,d)$ be a metric space and $f:X\to X$ be a homeomorphism. We say that a dynamical system $(X,f)$ is \emph{expansive}, with constant of expansivity $c\in \mathbb{R}{}^{+}$, if for all $x,y\in X$ , $x\ne y$, exists $n\in \mathbb{Z}$, such that $d(f^n(x),f^n(y))>c$. In this paper we will use the theory of Nonstandard Analysis to study a subfamily of these dynamics, which verify that for all $x,y\in X$, if $x\ne y$ then the set $\{n\in \mathbb{Z}:d(f^n(x),f^n(y)>c\}$ is infinite.