Homotopy shadowing

TL;DR

This paper generalizes Shub's 1969 result by extending the homotopy shadowing concept to certain expanding maps on metric spaces and hyperbolic maps on product manifolds, establishing a shadowing theorem with homotopy data.

Contribution

It introduces a homotopy shadowing theorem applicable to broader classes of maps, including expanding and hyperbolic maps, extending topological conjugacy classifications.

Findings

Established a shadowing theorem for pseudo-orbits with homotopy information

Generalized topological conjugacy results to metric spaces and product manifolds

Extended the understanding of homotopy types in dynamical systems

Abstract

Michael Shub proved in 1969 that the topological conjugacy class of an expanding endomorphism on a compact manifold is determined by its homotopy type. In this article we generalize this result in two directions. In one direction we consider certain expanding maps on metric spaces. In a second direction we consider maps which are hyperbolic with respect to product cone fields on a product manifold. A key step in the proof is to establish a shadowing theorem for pseudo--orbits with some additional homotopy information.