Generalized nonautonomous dynamics through groupoid morphisms

TL;DR

This paper generalizes nonautonomous dynamics using groupoid morphisms, introducing cotranslations and their applications to difference and differential equations, expanding the theoretical framework of dynamical systems.

Contribution

It introduces cotranslations as a new class of groupoid morphisms and establishes their connection to skew-products, broadening the scope of nonautonomous dynamics.

Findings

Cotranslations correspond to skew-products.

Applications to nonautonomous difference and differential equations.

Results on differentiability, dimension invariance, and diagonalization.

Abstract

We extend the notions of nonautonomous dynamics to arbitrary groups, through groupoid morphisms. This also presents a generalization of classic dynamical systems and group actions. We introduce the structure of cotranslations, as a specific kind of groupoid morphism, and establish a correspondence between cotranslations and skew-products. We give applications of cotranslations to nonautonomous equations, both in differences and differential. Our results delve into the differentiability of cotranslations, along with dimension invariance and diagonalization, utilizing a generalized notion of kinematic similarity.