Shadowing in the hyperspace of continua

TL;DR

This paper investigates whether classical dynamical systems with shadowing also exhibit shadowing on the hyperspace of continua, revealing both cases where it does and does not, with implications for dendrite maps and universal dendrites.

Contribution

It provides new results on shadowing properties for induced hyperspace maps, including conditions for dendrite maps and examples of systems with shadowing on complex spaces.

Findings

Transitive Anosov diffeomorphisms do not satisfy hyperspace shadowing.

Dendrite monotone maps satisfy hyperspace shadowing iff their induced maps do.

Universal dendrite of order n admits a homeomorphism with shadowing property.

Abstract

We discuss whether classical examples of dynamical systems satisfying the shadowing property also satisfy the shadowing property for the induced map on the hyperspace of continua, obtaining both positive and negative results. We prove that transitive Anosov diffeomorphisms, or more generally continuum-wise hyperbolic homeomorphisms, do not satisfy the shadowing property for the induced map on the hyperspace of continua. We prove that dendrite monotone maps satisfy the shadowing property if, and only if, their induced map on the hyperspace of continua also satisfies it. We give some algorithms that show that there are abundant dynamical systems satisfying the shadowing property with dendrites (compact metric trees) as the underlying space. As a consequence, we show that the universal dendrite of order n admits a homeomorphism with the shadowing property.