Algebraic entropy of amenable group actions

Virili, Simone

doi:10.1007/s00209-018-2192-0

Cited by 21 publications

(37 citation statements)

References 27 publications

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“…The same connection was given by Peters in [26] for topological automorphisms of metrizable compact abelian groups; moreover, these results were recently extended to continuous endomorphisms of compact abelian groups in [5], to continuous endomorphisms of totally disconnected locally compact abelian groups in [8], and to topological automorphisms of locally compact abelian groups in [34] (in a much more general setting). The problem of the validity of the Bridge Theorem in the general case of continuous endomorphisms of locally compact abelian groups is still open.…”

Section: Introductionmentioning

confidence: 56%

Topological entropy for locally linearly compact vector spaces

Castellano

Bruno

2019

Topology and its Applications

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In analogy to the topological entropy for continuous endomorphisms of totally disconnected locally compact groups, we introduce a notion of topological entropy for continuous endomorphisms of locally linearly compact vector spaces. We study the fundamental properties of this entropy and we prove the Addition Theorem, showing that the topological entropy is additive with respect to short exact sequences. By means of Lefschetz Duality, we connect the topological entropy to the algebraic entropy in a Bridge Theorem. 2010 MSC: Primary: 15A03; 15A04; 22B05. Secondary: 20K30; 37A35.

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Section: Introductionmentioning

confidence: 56%

Topological entropy for locally linearly compact vector spaces

Castellano

Bruno

2019

Topology and its Applications

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“…Furthermore, the following result for topological automorphisms of locally compact abelian groups was stated in [91], but several gaps in the proof were pointed out in [51]. It is now a consequence of a much more general result from [108]. Theorem 1.5.…”

Section: Bridge Theoremmentioning

confidence: 99%

“…In particular, the idea to use semigroups in place of semilattices belongs to him. He is presenting his own version of this general approach in [108]. Thanks are due also to George Janelidze for fruitful discussions with the first named author and in particular for his suggestion to introduce frame entropy.…”

Section: Acknowledgementsmentioning

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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“…They extend respectively the algebraic entropy ent introduced by Weiss [77] for endomorphisms of torsion abelian groups and the algebraic entropy h alg introduced in [28] following the work of Peters [64] for endomorphisms of abelian groups. For amenable group actions on discrete abelian groups our definition of algebraic entropy coincides with that given in [74] for locally compact abelian groups.…”

Section: Algebraic Entropy For Amenable Semigroup Actionsmentioning

confidence: 99%

“…The Bridge Theorem from [64] was recently extended by Virili [74] to the case of actions of amenable groups on locally compact abelian groups, while the one from [27] was extended in [39] to semigroup actions on totally disconnected locally compact abelian groups. In §7, generalizing the main result of [77], we prove a Bridge Theorem for left actions of cancellative left amenable monoids on totally disconnected compact abelian groups (their Pontryagin dual groups are precisely the torsion abelian groups).…”

Section: Introductionmentioning

confidence: 99%

Algebraic entropy for amenable semigroup actions

Dikranjan¹,

Fornasiero²,

Bruno³

2020

Journal of Algebra

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We introduce two notions of algebraic entropy for actions of cancellative right amenable semigroups S on discrete abelian groups A by endomorphisms; these extend the classical algebraic entropy for endomorphisms of abelian groups, corresponding to the case S = N. We investigate the fundamental properties of the algebraic entropy and compute it in several examples, paying special attention to the case when S is an amenable group.For actions of cancellative right amenable monoids on torsion abelian groups, we prove the so called Addition Theorem. In the same setting, we see that a Bridge Theorem connects the algebraic entropy with the topological entropy of the dual action by means of the Pontryagin duality, so that we derive an Addition Theorem for the topological entropy of actions of cancellative left amenable monoids on totally disconnected compact abelian groups.In Example 2.28 we show that the naïve generalization of this lemma for semidirect products fails, while Theorem 2.27 gives the correct generalization.Recall that a semigroup S is called Ore semigroup (or right reversible semigroup as in [63]) if it satisfies the left Ore condition, that is, for every a, b ∈ S there exist f, g ∈ S such that f a = gb.It is a well-known fact (see [63, Proposition 1.23]) that a right amenable semigroup S satisfies the left Ore condition, and that the group generated by S is of the form S −1 S.We use the following consequence of this property.Corollary 2.10. Let S be a right amenable semigroup, and s 1 , . . . , s k ∈ S. Then there exist t, r 1 , . . . , r k ∈ S such that t = r i s i for every i ∈ {1, . . . , k}.We conclude with the following relation between amenable monoids and amenable groups.

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Algebraic entropy of amenable group actions

Cited by 21 publications

References 27 publications

Topological entropy for locally linearly compact vector spaces

Topological entropy for locally linearly compact vector spaces

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

Algebraic entropy for amenable semigroup actions

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