New and old facts about entropy in uniform spaces and topological groups

Dikranjan, Dikran; Sanchis, Manuel; Virili, Simone

doi:10.1016/j.topol.2011.05.046

Cited by 42 publications

(49 citation statements)

References 25 publications

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“…The following basic case of Addition Theorem was already proved in [13,Corollary 4.17] in the general case of all topological groups, here we give a short proof for the case of locally compact groups. Next we give a useful application of Lemma 3.3.…”

Section: Addition Theoremmentioning

confidence: 75%

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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Section: Addition Theoremmentioning

confidence: 75%

Finiteness of Topological Entropy for Locally Compact Abelian Groups

Dikranjan¹,

Bruno²,

Russo³

2020

Glasgow Math. J.

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“…K-space and B(V ) denote the set of all linearly compact open K-subspaces of V . Inspired by the notion of topological entropy for totally disconnected locally compact groups and their endomorphisms (see [13,15]), the topological entropy for l.l.c. K-spaces is defined to be…”

Section: Topological Entropy and Algebraic Entropy Over K Llcmentioning

confidence: 99%

The corank of a flow over the category of linearly compact vector spaces

Castellano

2020

Journal of Pure and Applied Algebra

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For a topological flow (V, φ) -i.e., V is a linearly compact vector space and φ a continuous endomorphism of V -we gain a deep understanding of the relationship between (V, φ) and the Bernoulli shift: a topological flow (V, φ) is essentially a product of one-dimensional left Bernoulli shifts as many as ent * (V, φ) counts. This novel comprehension brings us to introduce a notion of corank for topological flows designed for coinciding with the value of the topological entropy of (V, φ). As an application, we provide an alternative proof of the so-called Bridge Theorem for locally linearly compact vector spaces connecting the topological entropy to the algebraic entropy by means of Lefschetz duality.

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“…Later on, Hood [76] extended Bowen-Dinaburg's entropy to uniformly continuous selfmaps of uniform spaces. This notion of entropy is sometimes called uniform entropy, and it coincides with the topological entropy in the compact case (when the given compact topological space is endowed with the unique uniformity compatible with the topology, see [48,51] for more detail). In particular, this topological entropy can be studied for continuous endomorphisms of topological groups.…”

Section: Historical Backgroundmentioning

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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New and old facts about entropy in uniform spaces and topological groups

Cited by 42 publications

References 25 publications

Finiteness of Topological Entropy for Locally Compact Abelian Groups

Finiteness of Topological Entropy for Locally Compact Abelian Groups

The corank of a flow over the category of linearly compact vector spaces

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

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