Algebraic entropy for amenable semigroup actions

Dikranjan, Dikran; Fornasiero, Antongiulio; Bruno, Anna Giordano

doi:10.1016/j.jalgebra.2020.02.033

Cited by 13 publications

(37 citation statements)

References 58 publications

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“…The same occurs for the two-sided right Bernoulli shiftβ ⊕ K in (5.11). It is proved in [32], in the more general case of amenable semigroup actions, that, for K an abelian group and λ : X → X a selfmap, h alg (τ λ ) = h(λ) · log |K|, with the convention that log |K| = ∞ if K is infinite. This generalizes the formula (5.12) for the right Bernoulli shifts and it will follow from Lemma 6.38 below.…”

Section: Algebraic Entropymentioning

confidence: 99%

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

Dikranjan

Bruno

2019

Dissertationes Math.

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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).

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Section: Algebraic Entropymentioning

confidence: 99%

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

Dikranjan

Bruno

2019

Dissertationes Math.

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“…which is called the n+ -coarse trajectory Tn(f , x, E) with respect to x and E. When there is no risk of ambiguity, we will simply call it n + trajectory. Note that, if X is locally nite, for every bornologous self-map, every trajectory is a nite subset according to (6). Before de ning the coarse entropy, in the notation above, let us focus a bit more on the entourages of the form (f n × f n )(E).…”

Section: De Nition Of Coarse Entropymentioning

confidence: 99%

“…More recently the entropy generated by actions of amenable semigroups and groups have been studied. In particular, Ornstein and Weiss introduced topological and measure entropy of amenable group actions ( [30]), whose approach was extended to the case of actions of amenable cancellative semigroups by Ceccherini-Silberstein, Coornaert and Krieger ( [5]), Hofmann and Stoyanov studied topological entropy of locally compact semigroup actions on metric spaces ( [21]), and Dikranjan, Fornasiero and Giordano Bruno in [6] de ned and discussed the algebraic entropy of an action of an amenable cancellative semigroup on an abelian group.…”

Section: Introductionmentioning

confidence: 99%

On a notion of entropy in coarse geometry

Zava

2019

Topological Algebra and Its Applications

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The notion of entropy appears in many branches of mathematics. In each setting (e.g., probability spaces, sets, topological spaces) entropy is a non-negative real-valued function measuring the randomness and disorder that a self-morphism creates. In this paper we propose a notion of entropy, called coarse entropy, in coarse geometry, which is the study of large-scale properties of spaces. Coarse entropy is defined on every bornologous self-map of a locally finite quasi-coarse space (a recent generalisation of the notion of coarse space, introduced by Roe). In this paper we describe this new concept, providing basic properties, examples and comparisons with other entropies, in particular with the algebraic entropy of endomorphisms of monoids.

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“…This result was extended to every abelian group in [11], again making a very heavy use of the properties of abelian groups (in order to reduce to the case of finite-dimensional rational vector spaces, where one applies the so-called algebraic Yuzvinski formula -see Remark 1.7 (b)). Recently, those results were generalized for a left action˛of a cancellative right amenable monoid S on an abelian group G (see Remark 1.7 (c)), in case G is torsion in [4], and when S is locally monotileable in [5].…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, in Section 3, we use the auxiliary function`. ; /, which was introduced in [4] in a much more general context. We consider this to be of independent interest.…”

mentioning

confidence: 99%

Additivity of the algebraic entropy for locally finite groups with permutable finite subgroups

Bruno

Flavio

2020

Journal of Group Theory

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Additivity with respect to exact sequences is, notoriously, a fundamental property of the algebraic entropy of group endomorphisms. It was proved for abelian groups by using the structure theorems for such groups in an essential way. On the other hand, a solvable counterexample was recently found, showing that it does not hold in general. Nevertheless, we give a rather short proof of the additivity of algebraic entropy for locally finite groups that are either quasihamiltonian or FC-groups.

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Algebraic entropy for amenable semigroup actions

Cited by 13 publications

References 58 publications

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

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

On a notion of entropy in coarse geometry

Additivity of the algebraic entropy for locally finite groups with permutable finite subgroups

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