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

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.