The categorical basis of dynamical entropy

TL;DR

This paper introduces a category-theoretic framework for understanding topological entropy in dynamical systems, showing how various complexity measures converge to a common limit through functorial relationships.

Contribution

It develops the concept of qualifying pairs of functors to explain the convergence of different complexity notions to topological entropy.

Findings

Diameter and Lebesgue number form a qualifying pair of functors.

Natural transformations lead to convergence of complexity measures.

Framework reveals structural reasons for the common limit in entropy notions.

Abstract

Many branches of theoretical and applied mathematics require a quantifiable notion of complexity. One such circumstance is a topological dynamical system - which involves a continuous self-map on a metric space. There are many notions of complexity one can assign to the repeated iterations of the map. One of the foundational discoveries of dynamical systems theory is that these have a common limit, known as the topological entropy of the system. We present a category-theoretic view of topological dynamical entropy, which reveals that the common limit is a consequence of the structural assumptions on these notions. One of the key tools developed is that of a qualifying pair of functors, which ensure a limit preserving property in a manner similar to the sandwiching theorem from Real Analysis. It is shown that the diameter and Lebesgue number of open covers of a compact space, form a qualifying pair of functors. The various notions of complexity are expressed as functors, and natural transformations between these functors lead to their joint convergence to the common limit.