Generalized dichotomies via time rescaling

TL;DR

This paper characterizes generalized dichotomies in discrete-time linear dynamics through time rescaling, extending previous polynomial cases, and applies these results to spectral analysis and linearization of nonautonomous systems.

Contribution

It provides a complete characterization of $$ dichotomies via time rescaling, generalizing known polynomial dichotomy results and applying them to spectral and linearization problems.

Findings

Characterization of $$ dichotomies via time rescaling.

Extension of polynomial dichotomy results to a broader class.

Application to the structure of the generalized Sacker-Sell spectrum.

Abstract

For discrete-time nonautonomous linear dynamics and a large class of discrete growth rates $\mu$, we show that the notion of $\mu$ dichotomy (with respect to a sequence of norms) can be completely characterized in terms of ordinary and exponential dichotomy (with respect to a sequence of norms) by employing a suitable rescaling of time. Previously, such a result was known only in the particular case of polynomial dichotomies. As a nontrivial application of our results, we study the structure of a generalized Sacker-Sell spectrum and obtain a series of nonautonomous topological and smooth linearization results.