A note on the differentiability of Palmer's topological equivalence for discrete systems

TL;DR

This paper investigates conditions under which topological equivalence between linear and nonlinear discrete systems is differentiable, focusing on dichotomy conditions and perturbation bounds to establish $C^1$ smoothness.

Contribution

It provides sufficient conditions for the homeomorphism of topological equivalence to be a $C^1$ diffeomorphism in discrete systems with dichotomies.

Findings

Homeomorphism can be a $C^1$ diffeomorphism under certain conditions.

Dichotomy conditions like exponential and nonuniform exponential are considered.

Potential extension to higher differentiability classes discussed.

Abstract

A linear system of difference equations and a nonlinear perturbation are considered. We propose sufficient conditions to ensure that the homeomorphism of topological equivalence between them is actually a $C^1$ diffeomorphism. These conditions consider that the linear part satisfies a dichotomy on the positive half-line, while the perturbation satisfies boundness and Lipschitzness conditions, and anothers that are tailored for our goal. The family of dichotomies that we study consider (but are not limited to) the exponential and nonuniform exponential dichotomies. We discuss the possibility of extending this result to a higher class of differentiability.