Entropy of automorphisms on L.C.A. groups

Peters, Justin R.

doi:10.2140/pjm.1981.96.475

Cited by 34 publications

(31 citation statements)

References 10 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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“…However the validity of the Addition Theorem in the general case of locally compact groups and their continuous endomorphisms is not yet established even in the abelian setting, to the best of our knowledge. (A wrong proof of the Addition Theorem for locally compact abelian groups appeared in [25] -see [13] for more detail. )…”

Section: Then Also the Converse Implications Hold In (A) And (B)mentioning

confidence: 99%

Finiteness of Topological Entropy for Locally Compact Abelian Groups

Dikranjan¹,

Bruno²,

Russo³

2020

Glasgow Math. J.

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We study the locally compact abelian groups in the class E<∞, that is, having only continuous endomorphisms of finite topological entropy, and in its subclass E 0 , that is, having all continuous endomorphisms with vanishing topological entropy. We discuss the reduction of the problem to the case of periodic locally compact abelian groups, and then to locally compact abelian p-groups.We show that locally compact abelian p-groups of finite rank belong to E<∞; moreover, those of them that belong to E 0 are precisely the ones with discrete maximal divisible subgroup. Furthermore, the topological entropy of endomorphisms of locally compact abelian p-groups of finite rank coincides with the logarithm of their scale.The backbone of the paper is the Addition Theorem for continuous endomorphisms of locally compact abelian groups. Various versions of the Addition Theorem are established in the paper and used in the proofs of the main results, but its validity in the general case remains an open problem.

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“…Peters [91] gave a further generalization of his notion of entropy h alg for continuous automorphisms of locally compact abelian groups, which was recently extended by Virili [107] to continuous endomorphisms; the commutativity can be removed as noted in [36]. The recent papers [62,115] are dedicated to the study of the algebraic entropy for some classes of locally compact not necessarily abelian groups where the Addition Theorem is available.…”

Section: Historical Backgroundmentioning

confidence: 99%

“…It turns out that the value (What Peters defined actually was, in our notation, h alg (φ −1 ) for a topological automorphism φ; it is denoted by h ∞ (φ) in [91].) As we saw above (7.2) is a limit when G is discrete.…”

Section: Topological and Algebraic Entropy In Locally Compact Groupsmentioning

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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Entropy of automorphisms on L.C.A. groups

Cited by 34 publications

References 10 publications

Intrinsic algebraic entropy

Intrinsic algebraic entropy

Finiteness of Topological Entropy for Locally Compact Abelian Groups

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

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