Groupoid morphisms as an algebraic structure for nonautonomous dynamics

TL;DR

This paper introduces groupoid morphisms as a new algebraic framework for nonautonomous dynamics, generalizing classical group morphisms and applying to difference and differential equations with properties like differentiability and dimension invariance.

Contribution

It develops the concept of cotranslations as a specific type of groupoid morphism and links them to skew-products, expanding the tools for analyzing nonautonomous systems.

Findings

Cotranslations correspond to skew-products in nonautonomous dynamics.

Results on differentiability and dimension invariance of cotranslations.

Extension of kinematic similarity to noninvertible transformations.

Abstract

We present groupoid morphisms as an algebraic structure for nonautonomous dynamics, as well as a generalization of group morphisms, which describe classic dynamical systems. 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. We obtain results about the differentiability of cotranslations, as well as dimension invariance and diagonalization (through a generalized notion of kinematic similarity) for a partial version of them, admitting noninvertible transformations.