Periodic Shadowing for Set-Valued Maps
M. Oliveira
TL;DR
This work extends shadowing theory to set-valued dynamics by introducing and analyzing periodic shadowing ($PeSP$). It proves that, on compact spaces, positive expansivity plus shadowing ($SP$) forces $PeSP$, while chain transitivity plus $PeSP$ implies both $SP$ and topological transitivity, clarifying how nondeterministic dynamics relate to classical notions. A general construction transfers $SP$ and $PeSP$ from a single-valued system to a set-valued one via an isometric involution, and the results are reinforced with symbolic dynamics examples. The findings provide stability and compositional properties for $PeSP$ (invariance under iterations, inverses, and products) and offer practical tools for building robust shadowing behavior in set-valued models.
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.