h-Dichotomies via noncritical uniformity and expansiveness for evolution families

TL;DR

This paper extends the characterization of h-dichotomies from finite-dimensional differential equations to infinite-dimensional Banach space evolution families, using a novel, simpler approach based on time-rescaling.

Contribution

It generalizes existing finite-dimensional results to Banach spaces with a new, simpler method based on time-rescaling, expanding the applicability of h-dichotomy theory.

Findings

Extended h-dichotomy characterization to Banach space evolution families.

Introduced a new approach based on time-rescaling for analyzing dichotomies.

Simplified the proof technique compared to previous methods.

Abstract

In a recent paper (Math. Ann. 393 (2025), 1769--1795), Elorreaga et al. have obtained a complete characterization of the notion of a $h$-dichotomy for ordinary differential equations on a finite-dimensional space in terms of the notions of $h$-expansiveness and $h$-noncriticality. Their results extended the previous results of Coppel and Palmer, which dealt with exponential dichotomies. The main objective of this note is to extend the results of Elorreaga et al. to arbitrary invertible evolution families that act on Banach spaces. We emphasize that our approach is completely different and considerably simpler from the one developed by Elorreaga et al. It is based on the time-rescaling method introduced by Dragicevic and Silva.