Nonstandard Expansiveness

TL;DR

This paper employs Nonstandard Analysis to explore a special class of expansive dynamical systems where pairs of distinct points repeatedly separate beyond a fixed distance infinitely often.

Contribution

It introduces a nonstandard analytical approach to study a subclass of expansive systems with infinite recurrence of separation.

Findings

Identifies a new class of expansive systems with infinite separation recurrence.

Uses nonstandard analysis to characterize these systems.

Provides theoretical insights into their dynamical behavior.

Abstract

Let $(X,d)$ be a metric space and $f: X \rightarrow X$ 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 $x,y \in X$ , $x \neq y$, 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 $x\neq y$ then the set $\lbrace n \in \mathbb{Z} : d(f^n(x), f^n(y) > c \rbrace$ is infinite.