On Stronger Forms of Expansivity

TL;DR

This paper introduces stronger forms of positively expansive maps called $p \\mathscr{F}-$expansive maps, explores their properties, examples, and conditions, and studies their behavior in various dynamical systems contexts.

Contribution

It defines and analyzes $p \\mathscr{F}-$expansive maps, providing new examples, characterizations, and studying their properties and limitations in dynamical systems.

Findings

Constructed examples of positively thick and syndetic expansive maps.

Established conditions under which positively expansive maps are co-finite or syndetic expansive.

Proved non-existence of certain expansive homeomorphisms on compact metric spaces.

Abstract

We define the concept of stronger forms of positively expansive map and name it as $p \:\mathscr{F}-$expansive maps. Here $\mathscr{F}$ is a family of subsets of $\mathbb{N}$. Examples of positively thick expansive and positively syndetic expansive maps are constructed here. Also, we obtain conditions under which a positively expansive map is positively co--finite expansive and positively syndetic expansive maps. Further, we study several properties of $p \:\mathscr{F}-$expansive maps. A characterization of $p \:\mathscr{F}-$expansive maps in terms of $p \:\mathscr{F}^*-$generator is obtained. Here $p \:\mathscr{F}^*$ is dual of $\mathscr{F}$. Considering $(\mathbb{Z},+)$ as a semigroup, we study $\mathscr{F}-$expansive homeomorphism, where $\mathscr{F}$ is a family of subsets of $\mathbb{Z} \setminus \{0\}$. We show that there does not exists an expansive homeomorphism on a compact metric space which is $\mathscr{F}_s-$expansive. Also, we study relation between $\mathscr{F}-$expansivity of $f$ and the shift map $\sigma_f$ on the inverse limit space.