## Entropy on normed semigroups (towards a unifying approach to entropy)

### Volume 542 / 2019

#### Abstract

We present a unifying approach to the study of entropies in mathematics, such as measure entropy, various forms of topological entropy, several notions of algebraic entropy, and two forms of set-theoretic entropy. We take into account only discrete dynamical systems, that is, pairs $(X,\phi)$, where $X$ is the underlying space (e.g., a probability space, a compact topological space, a group, a set) and $\phi:X\to X$ is a transformation of $X$ (e.g., a measure preserving transformation, a continuous selfmap, a group homomorphism, a selfmap). We see entropies as functions $h:\mathfrak X\to \mathbb R_+$, associating to each flow $(X,\phi)$ of a category $\mathfrak X$ either a non-negative real number or $\infty$.

First, 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 ${\bf h}_F:\mathfrak X\to\mathbb R_+$ as the composition $h_{\mathfrak S}\circ F$, that is, ${\bf h}_F(\phi) = h_{\mathfrak S}(F\phi)$ for any endomorphism $\phi: X \to X$ in $\mathfrak X$. Clearly, ${\bf h}_F$ inherits many of the properties of $h_{\mathfrak S}$, depending also on the functor $F$. Motivated by this aspect, we study in detail the properties of $h_{\mathfrak S}$.

Such a general scheme, using elementary category theory, permits one to obtain many relevant known entropies as functorial entropies ${\bf h}_F$, for appropriately chosen categories $\mathfrak X$ and functors $F:\mathfrak X\to{\mathfrak S}$. All of the above mentioned entropies are functorial. Furthermore, we exploit our scheme to elaborate a common approach to establishing the properties shared by those entropies that we find as functorial entropies, pointing out their common nature. We give also a detailed description of the limits of our approach, namely entropies which cannot be covered.

Finally, we discuss and deeply analyze the relations between pairs of entropies through the looking glass of our unifying approach. To this end we first formalize the notion of Bridge Theorem between two entropies $h_1:\mathfrak X_1\to \mathbb R_+$ and $h_2:\mathfrak X_2\to \mathbb R_+$ with respect to a functor $\varepsilon:\mathfrak X_1\to\mathfrak X_2$, taking inspiration from the known relation between the topological and the algebraic entropy via the Pontryagin duality functor. Then, for pairs of functorial entropies we use the above scheme to introduce the notion and the related scheme of Strong Bridge Theorem. It allows us to shelter various relations between pairs of entropies under the same umbrella (e.g., the above mentioned connection of the topological and the algebraic entropy, as well as their relation to the set-theoretic entropy).