Work

Shadowing in the hyperspace of continua

Year: 2025

Type: article

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.

Source: Nonlinearity

DOI: 10.1088/1361-6544/adfff5

Authors: Bernardo Carvalho, Udayan Darji

Institutions: University of Rome Tor Vergata, Laboratório Nacional de Computação Científica, University of Louisville, Charles University

Language: English

Topic: Mathematical Dynamics and Fractals

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,325

Funder: Università degli Studi di Roma Tor Vergata