Periodic Shadowing for Set-Valued Maps

TL;DR

This paper explores shadowing properties in set-valued dynamical systems, establishing key relationships between periodic shadowing, expansivity, and chain transitivity, with applications to symbolic dynamics.

Contribution

It introduces new results linking periodic shadowing with expansivity and chain transitivity in set-valued maps, and constructs examples from single-valued systems with involutions.

Findings

Positively expansive set-valued maps on compact spaces have shadowing imply periodic shadowing.

Periodic shadowing in chain transitive maps implies shadowing and topological transitivity.

Examples include systems from symbolic dynamics illustrating the theory.

Abstract

We study shadowing-type properties for set-valued dynamical systems. In particular, we investigate the periodic shadowing property and its relationship with expansivity and chain transitivity. We establish that for positively expansive set-valued maps on compact metric spaces, the shadowing property implies the periodic shadowing property. Furthermore, we show that for chain transitive maps, periodic shadowing implies both shadowing and topological transitivity. We also present a general construction of set-valued maps with shadowing arising from single-valued systems admitting an isometric involution. Several examples, including systems from symbolic dynamics, are provided to illustrate the theory.