Entropy for endomorphisms of LCA groups

Virili, Simone

doi:10.1016/j.topol.2011.02.017

Cited by 34 publications

(41 citation statements)

References 7 publications

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“…This notion was extended to endomorphisms of Abelian groups in [6], where it was proved that h coincides with the algebraic entropy ent for endomorphisms of torsion Abelian groups. Imitating [16] and [6], in [23] the definition of algebraic entropy ✩ This research was partially supported by "Progetti di Eccellenza 2011/12" of Fondazione CARIPARO. The fourth named author was partially supported by DGI MINECO MTM2011-28992-C02-01, and by the Comissionat per Universitats i Recerca de la Generalitat de Catalunya.…”

Section: Introductionmentioning

confidence: 99%

Intrinsic algebraic entropy

Dikranjan

Bruno

Salce

et al. 2015

Journal of Pure and Applied Algebra

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The new notion of intrinsic algebraic entropy (ent) over tilde of endomorphisms of Abelian groups is introduced and investigated. The intrinsic algebraic entropy is compared with the algebraic entropy, a well-known numerical invariant introduced in the sixties and recently deeply studied also in its relations to other fields of Mathematics. In particular, it is shown that the intrinsic algebraic entropy and the algebraic entropy coincide on endomorphisms of torsion Abelian groups, and their precise relation is clarified in the torsion-free case. The Addition Theorem and the Uniqueness Theorem are also proved for ent, in analogy with similar results for the algebraic entropy. Furthermore, a relevant connection of eat to the algebraic entropy of a continuous endomorphism of a locally compact Abelian group G is pointed out; this allows for the computation of the algebraic entropy in case G is totally disconnected

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Section: Introductionmentioning

confidence: 99%

Intrinsic algebraic entropy

Dikranjan

Bruno

Salce

et al. 2015

Journal of Pure and Applied Algebra

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“…Note that these two definitions coincide on automorphisms of torsion abelian groups. Throughout the years there have been several extensions to Peters' entropy (see [7,20,24]).…”

Section: Introductionmentioning

confidence: 99%

“…Following Virili [24] (see also [4]) we give now the general definition for the algebraic entropy on (not necessarily abelian) locally compact groups. Let G be a locally compact group and µ be a right Haar measure on G. For φ ∈ End(G), a subset U ⊆ G, and n ∈ N + , the n-th φ-trajectory of U is…”

Section: Introductionmentioning

confidence: 99%

Algebraic entropy on topologically quasihamiltonian groups

Shlossberg

Toller

2020

Topology and its Applications

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We study the algebraic entropy of continuous endomorphisms of compactly covered, locally compact, topologically quasihamiltonian groups. We provide a Limit-free formula which helps us to simplify the computations of this entropy. Moreover, several Addition Theorems are given. In particular, we prove that the Addition Theorem holds for endomorphisms of quasihamiltonian torsion FC-groups (e.g., Hamiltonian groups).2010 Mathematics Subject Classification. 37A35, 22D40, 28D20, 20K35.

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“…Several different notions of algebraic entropy have been introduced in the past (see [1,11,17,14] and references there). In particular, the possibility to define φ-inert subobjects has recently turned out to be a very helpful tool for the study of the dynamical properties of the given endomorphism φ.…”

Section: Connection With Algebraic Entropymentioning

confidence: 99%

A property of the lamplighter group

Castellano

Cook

Kropholler

2020

Topological Algebra and Its Applications

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We show that the inert subgroups of the lamplighter group fall into exactly five commensurability classes. The result is then connected with the theory of totally disconnected locally compact groups and with algebraic entropy.A subgroup H of a group G is said to be inert if H and g −1 Hg are commensurate for all g ∈ G, meaning that H ∩ H g always has finite index in both H and H g . The terminology was introduced by Kegel and has been explored in many contexts (see, for example, the recent survey [8]). In abstract group theory, Robinson's investigation [13] focusses on soluble groups. Here we study a particular special family of soluble groups: the lamplighter groups and our interest is in the connection with the theory of totally disconnected locally compact groups. In that context, inert subgroups are particularly important in the light of van Dantzig's theorem that every totally disconnected locally compact group has a compact open subgroup and of course all such subgroups are commensurate with one another and therefore inert. It should be noted that in recent literature it is common to use the term commensurated in place of inert, see for example [5,6,7,10]. In §2 we remark a dynamical aspect of the property investigated here and relate it to the concept of algebraic entropy.The relation of commensurability is an equivalence relation amongst the subgroups of a group. By a class we shall here mean an equivalence class of subgroups under this relation. For a prime p, the corresponding lamplighter group is the standard restricted wreath product F p wr Z, i.e., the standard restricted wreath product of a group of order p by an infinite cyclic group. These are the simplest of soluble groups that fall outside the classes considered by Robinson [13]. Our main observation is as follows.Theorem. The inert subgroups of the lamplighter group F p wr Z fall into exactly five classes.

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Entropy for endomorphisms of LCA groups

Cited by 34 publications

References 7 publications

Intrinsic algebraic entropy

Intrinsic algebraic entropy

Algebraic entropy on topologically quasihamiltonian groups

A property of the lamplighter group

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