Homotopy shadowing

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.

arXiv comment42 pages, 17 figures. This version contains a corrected proof of Proposition 5.11