Uniform $h$-dichotomies: noncritical uniformity and expansivity

TL;DR

This paper generalizes the concept of exponential dichotomy to uniform $h$-dichotomies, establishing equivalences with noncritical uniformity and expansivity using ordered topological groups, extending classical results in dynamical systems.

Contribution

It introduces a framework linking $h$-dichotomies with ordered topological groups, broadening the scope of exponential dichotomy theory in nonautonomous systems.

Findings

Generalization of exponential dichotomy to uniform $h$-dichotomies.

Establishment of equivalences with noncritical uniformity and expansivity.

Use of ordered topological groups to facilitate the generalization.

Abstract

The property of exponential dichotomy can be seen as a generalization of the hyperbolicity condition for non autonomous linear finite dimensional systems of ordinary differential equations. In 1978 W.A. Coppel proved that the exponential dichotomy on the half line is equivalent to the property of noncritical uniformity provided that a condition of bounded growth is verified. In 2006 K.J. Palmer extended this result by proving that -- also assuming the bounded growth property -- the exponential dichotomy on the half line, noncritical uniformity and the exponential expansiveness are equivalent. The main contribution of this article is to generalize these results for the property of uniform $h$-dichotomy. This has been carried out due to a recent idea: under suitable conditions any $h$-dichotomy can be associated to a totally ordered topological group, which becomes the additive group $(\mathbb{R},+)$ in case of the exponential dichotomy. The properties of this new group make possible such generalization.