Orbitwise expansive maps
Debasish Bhattacharjee, Humayan Kobir, Santanu Acharjee
TL;DR
This work introduces orbitwise expansive (OE) and relatively orbitwise expansive (ROE) notions as point- and set-level generalizations of classical expansiveness. By formalizing OE and ROE, it unifies and extends several existing expansiveness concepts (including expansive and CW-expansive maps) and establishes key properties such as preservation under unions, behavior at limit points, and product/conjugacy stability. The authors also extend OE/ROE to time-varying dynamical systems, showing OE/ROE behavior under time-dependent maps and providing illustrative examples where OE/ROE hold without full expansiveness. The results yield a flexible framework for studying chaotic behavior in both static and time-varying contexts, with several open questions about equivalences and implications in broader settings.
Abstract
This study defines an orbitwise expansive point (OE) as a point, such as $x$ in a metric space $(X,ρ)$, if there is a number $d>0$ such that the orbits of a few points inside an arbitrary open sphere will maintain a distance greater than $d$ from the corresponding points of the orbit of $x$ at least once. The point $x$ is referred to as the relatively orbitwise expansive point (ROE) in the previously described scenario if $d$ is replaced with the radius of the open sphere whose orbit is investigated and whose centre is $x$. %The function generating the orbit is considered to be continuous. We also define OE (ROE) set. We prove that arbitrary union of OE (ROE) set is again OE (ROE) set and every limit point of an OE set is an OE point. We show that, rather than the other way around, Utz's expansive map or Kato's CW-expansive map implies OE (ROE) map. We utilise the concept of OE(ROE) to analyse a time-varying dynamical system and investigate its relevance to certain traits associated with expansiveness.