Algebraic Entropy of Shift Endomorphisms on Abelian Groups

Akhavin, Maryam; Dikranjan, Dikran; Bruno, Anna Giordano; Hosseini, Arezoo; Shirazi, Fatemah Ayatollah Zadeh

doi:10.2989/qm.2009.32.4.3.961

Cited by 14 publications

(33 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In case λ is finitely many-to-one, the subgroup K (X) is σ λ -invariant, so one can consider also the generalized shift σ λ : K (X) → K (X) . In [3] and [24], h(σ λ ) was computed on direct sums and on direct products, respectively. Remark 2.11.…”

Section: Basic Properties Of Entropymentioning

confidence: 99%

Entropy on abelian groups

Dikranjan

Bruno

2016

Advances in Mathematics

Self Cite

View full text Add to dashboard Cite

We introduce the algebraic entropy for endomorphisms of arbitrary abelian groups, appropriately modifying existing notions of entropy. The basic properties of the algebraic entropy are given, as well as various examples. The main result of this paper is the Addition Theorem showing that the algebraic entropy is additive in appropriate sense with respect to invariant subgroups. We give several applications of the Addition Theorem, among them the Uniqueness Theorem for the algebraic entropy in the category of all abelian groups and their endomorphisms. Furthermore, we point out the delicate connection of the algebraic entropy with the Mahler measure and Lehmer Problem in Number Theory

show abstract

Section: Basic Properties Of Entropymentioning

confidence: 99%

Entropy on abelian groups

Dikranjan

Bruno

2016

Advances in Mathematics

Self Cite

View full text Add to dashboard Cite

show abstract

“…Then we discuss the contravariant set-theoretic entropy h * from [36]. These entropies h and h * are related to invariants for selfmaps of sets (i.e., the string number and the antistring number, see [3,45,58,65]).…”

Section: Set-theoretic Entropymentioning

confidence: 99%

“…In particular, when Y = X, then K (X) is a σ λ -invariant subgroup of K X precisely when λ is finite-toone. It is known from [36,Theorem 7.3.3] and [3] that, for the contravariant set-theoretic entropy h * p defined there (see Remark 5.4), (5.14) with the convention that log |K| = ∞ if K is infinite. As proved in [3], the formula in (5.12) can be deduced from (5.14).…”

Section: Algebraic Entropymentioning

confidence: 99%

“…It is known from [36,Theorem 7.3.3] and [3] that, for the contravariant set-theoretic entropy h * p defined there (see Remark 5.4), (5.14) with the convention that log |K| = ∞ if K is infinite. As proved in [3], the formula in (5.12) can be deduced from (5.14). Moreover, in the concrete category AG, the forward generalized shift from Definition 4.16 has the following description.…”

Section: Algebraic Entropymentioning

confidence: 99%

See 1 more Smart Citation

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

Dikranjan

Bruno

2019

Dissertationes Math.

Self Cite

View full text Add to dashboard Cite

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).

show abstract

“…These were generalized to all abelian groups in [23]. Details and results can be found in [2,4,23,29,38,42,43], in [40,41] for the non-abelian case, in [66,67] for the algebraic entropy for modules; see also the surveys [26,28,33,44,45].…”

Section: Introductionmentioning

confidence: 99%

Algebraic entropy for amenable semigroup actions

Dikranjan¹,

Fornasiero²,

Bruno³

2020

Journal of Algebra

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Algebraic Entropy of Shift Endomorphisms on Abelian Groups

Cited by 14 publications

References 5 publications

Entropy on abelian groups

Entropy on abelian groups

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

Algebraic entropy for amenable semigroup actions

Contact Info

Product

Resources

About