Categorical foundations of discrete dynamical systems

TL;DR

This paper develops a categorical framework for discrete dynamical systems, introducing cycle sets to formalize attractors and providing a decomposition theorem to analyze system structures and their dynamics.

Contribution

It introduces the concept of cycle sets as a new formal tool for understanding attractors within a categorical foundation for discrete dynamical systems.

Findings

Cycle sets effectively formalize attractors.

A decomposition theorem for discrete dynamical systems is established.

The framework links system structure to dynamic behavior.

Abstract

We develop categorical foundations of discrete dynamical systems, aimed at understanding how the structure of the system affects its dynamics. The key technical innovation is the notion of a cycle set, which provides a formal language in which to speak of the system's attractors. As a proof of concept, we provide a decomposition theorem for discrete dynamical systems.