A new approach to integral characterizations in dichotomy theory

Abstract

This paper introduces a new approach to integral characterizations in the study of dynamical systems, utilizing the connection between uniform dichotomy with differentiable growth rates and uniform exponential dichotomy. Within this framework, we establish integral conditions for these dichotomies, considering both invariant projection-valued function and projection-valued function compatible with a skew-evolution cocycle. The method relies on incorporating differentiable growth rates, which extend existing results and provide a broader perspective on dichotomy theory. We hope that these results may contribute to the dichotomy theory of nonautonomous dynamical systems.