Topological simplification of nonautonomous difference equations

Abstract

In this paper, we continue the study of geometric properties of nonautonomous difference equations in arbitrary Banach spaces which was begun in [2,3]. Building on previous results on invariant fiber bundles and foliations, this paper addresses the problem of topological simplifications via continuous conjugacies and semiconjugacies. In particular, we establish a reduction principle for not necessarily invertible difference equations, as well as a generalized Hartman–Grobman theorem for systems with not necessarily invertible linear part.

Keywords:

• Hartman–Grobman theorem
• Reduction principle
• Decoupling
• Nonautonomous difference equation
• Noninvertible difference equation

Keywords:

• 39A10
• 39A11

Acknowledgements

As was mentioned in the introduction, this paper is the last in a series of papers on the qualitative behavior of nonautonomous dynamical processes. These results were based on the theses [10,11] by the second author, for which Bernd Aulbach served as the advisor. While the main parts of the paper had basically been finished some time ago, Bernd Aulbach could not see the final version of this paper due to his sudden and unexpected death. Nevertheless, he was an integral part of the work leading to this series and is therefore listed as one of the authors.