Entropy on normed semigroups (Towards a unifying approach to entropy)

TL;DR

This paper introduces a unifying categorical framework for various types of entropy in mathematics, using normed semigroups and functorial constructions to relate and analyze different entropy notions systematically.

Contribution

It develops a general scheme to unify and analyze different entropy concepts via functorial and semigroup approaches, revealing their common structure and relations.

Findings

Introduces semigroup entropy as a numerical invariant for endomorphisms of normed semigroups.

Defines functorial entropy as a composition of semigroup entropy with category functors.

Establishes the concepts of Bridge and Strong Bridge Theorems to relate different entropy measures.

Abstract

We present a unifying approach to the study of entropies in Mathematics, such as measure entropy, topological entropy, algebraic entropy, set-theoretic entropy. We take into account discrete dynamical systems, that is, pairs $(X,T)$, where $X$ is the underlying space and $T:X\to X$ a transformation. We see entropies as functions $h:\mathfrak X\to \mathbb R_+$, associating to each flow $(X,T)$ of a category $\mathfrak X$ either a non negative real or $\infty$. We introduce the notion of semigroup entropy $h_\mathfrak S:\mathfrak S\to\mathbb R_+$, which is a numerical invariant attached to endomorphisms of the category $\mathfrak S$ of normed semigroups. Then, for a functor $F:\mathfrak X\to\mathfrak S$ from any specific category $\mathfrak X$ to $\mathfrak S$, we define the functorial entropy $h_F:\mathfrak X\to\mathbb R_+$ as the composition $h_{\mathfrak S}\circ F$. Clearly, $h_F$ inherits many of the properties of $h_\mathfrak S$, depending also on the properties of $F$. Such general scheme permits to obtain relevant known entropies as functorial entropies $h_F$, for appropriate categories $\mathfrak X$ and functors $F$, and to establish the properties shared by them. In this way we point out their common nature. Finally, we discuss and deeply analyze through the looking glass of our unifying approach the relations between pairs of entropies. To this end we formalize the notion of Bridge Theorem between two entropies $h_i:\mathfrak X_i\to \mathbb R_+$, $i=1,2$, with respect to a functor $\varepsilon:\mathfrak X_1\to\mathfrak X_2$. Then, for pairs of functorial entropies we use the above scheme to introduce the notion and the related scheme of Strong Bridge Theorem, which allows us to put under the same umbrella various relations between pairs of entropies.