Bi-Asymptotic $c$-Expansivity

TL;DR

This paper introduces bi-asymptotically $c$-expansive maps in metric spaces, explores their relation to other expansivity types, and proves a spectral decomposition theorem for such maps with shadowing.

Contribution

It defines a new class of maps called bi-asymptotically $c$-expansive and establishes their properties and relationship with existing expansivity concepts.

Findings

Bi-asymptotically $c$-expansive maps are distinct from other expansivity variants.

Expansive homeomorphisms are not necessarily bi-asymptotically expansive.

A spectral decomposition theorem is proved for bi-asymptotically $c$-expansive maps with shadowing.

Abstract

In this paper, we define bi-asymptotically $c$-expansive maps on metric spaces and study its relationship with other variants of expansivity such as bi-asymptotically expansive maps and $N$-expansive maps. We also provide an example to establish that expansive homeomorphisms need not be bi-asymptotically expansive. Finally we prove a spectral decomposition theorem for bi-asymptotically $c$-expansive continuous surjective maps with the shadowing property on compact metric spaces.