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, various forms of topological entropy, several notions of algebraic entropy, and two forms set-theoretic entropy. We take into account only discrete dynamical systems, that is, pairs (X, T ), where X is the underlying space (e.g., a probability space, a compact topological space, a group, a set) and T : X → 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 : X → R + , associating to each flow (X, φ) of a category X either a non negative real number or ∞.First, we introduce the notion of semigroup entropy h S : S → R + , which is a numerical invariant attached to endomorphisms of the category S of normed semigroups. Then, for a functor F : X → S from any specific category X to S, we define the functorial entropy h F :Clearly, this entropy h F inherits many of the properties of h S , depending also on the properties of the functor F . Motivated by this aspect, we study in detail the properties of h S . Such general scheme, using elementary category theory, permits to obtain many relevant known entropies as functorial entropies h F , for appropriately chosen categories X and functors F : X → S. For example, most of the above mentioned measure entropy, topological entropies, algebraic entropies and set-theoretic entropies are functorial entropies. Furthermore, we exploit our scheme to elaborate a common approach to establish the properties shared by those entropies that we find as functorial entropies. 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 first formalize the notion of Bridge Theorem between two entropies h 1 : X 1 → R + and h 2 : X 2 → R + with respect to a functor ε : X 1 → X 2 , taking inspiration from the theorem relating the topological and the algebraic entropy by means of Pontryagin duality. 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 put under the same umbrella various relations between pairs of entropies (e.g., the above mentioned connection of the topological and the algebraic entropy and their relation with the set-theoretic entropy).

Metric Versus Topological Receptive Entropy of Semigroup Actions

“…There is an obvious analogy between the Variational Principle, connecting (receptive) topological entropy and (receptive) metric entropy of the same action T , and the Bridge Theorem, connecting the (receptive) topological entropy of an action T and the (receptive) algebraic entropy of the dual action T (for example see [15]). This motivated our choice to formulate Conjecture 6.3 in view of the positive evidence, in this sense, provided by the Bridge Theorem available in all these cases, as well as the results of Sect.…”

On a notion of entropy in coarse geometry

“…Actually, we will show that hc(id X ) can only take values in { , ∞} (Theorem 4.4), and thus hc(id X ) = ∞. (b) Let us slightly modify item (a), by changing the sequence {Kn}n in (11) to…”

“…Other notions of topological entropy were given by Bowen ([4]) and Hood ([22]). In algebraic dynamics, we can cite the work of Adler, Konheim and MacAndrew ( [1]), the entropy de ned by Weiss in [43], and the one introduced by Peters ( [31], and deeply studied in [12]), that was then generalised in [10] for endomorphisms of abelian groups (we refer to [8] for the de nition in the non-abelian case, while to [11] for the extension to endomorphisms of semigroups). Later, Peters in [32] gave an extension of the algebraic entropy de ned in [31] for topological automorphisms of locally compact abelian groups.…”

“…This de nition can be found in [8] also for non-abelian groups. We nally mention the paper [11], where a unifying approach to several notions of entropy is provided by using normed semigroups. More recently the entropy generated by actions of amenable semigroups and groups have been studied.…”

Ore localization of amenable monoid actions and applications towards entropy $-$ addition formulas and the bridge theorem

“…The proof of Theorem A.24, as well as that of (at top ), is heavily based on the uniform approach to entropy via the entropy of monoid actions on normed monoids developed in [12,17] for N-actions and then extended to the general case in [57,58]. This approach covers, beyond the algebraic and the various versions of the topological entropy, also the measure entropy and many others (see [17]). It is exposed with more details in §1.4.…”