Topological entropy for locally linearly compact vector spaces

Castellano, Ilaria; Bruno, Anna Giordano

doi:10.1016/j.topol.2018.11.009

Cited by 11 publications

(24 citation statements)

References 37 publications

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“…2 One can check that the dual factorisation does always exist in the dual category K Vect eP4 (Continuity on inverse limits) Let {W i | i ∈ I} be a directed system (for inverse inclusion) of closed Notice that the topological entropy over K LLC was originally denoted in [5] simply by ent * . Here we need to keep in mind the field we are working on.…”

Section: Topological Entropy On K Llcmentioning

confidence: 99%

“…Not only because locally compact groups are omnipresent in several areas of mathematics but also because there exists a strict relation between the topological entropy h top and the scale function defined in [9] whenever one considers endomorphisms of totally disconnected locally compact groups (see [2,7] for details). By analogy with the topological entropy h top for totally disconnected locally compact groups, the topological entropy ent * has been introduced in the category K LLC of locally linearly compact K-spaces [5] whenever K is a discrete field. The main motivation for studying such an entropy function is to reach a better understanding of h top by means of the more rigid case of locally linearly compact spaces.…”

Section: Introductionmentioning

confidence: 99%

“…There are no surprises in either the statement or the proof of Theorem 4.4, but its content does not appear anywhere else. Thus this paper is intended to be just a complement to [4,5] where ent * for locally linearly compact K-spaces has been introduced and studied.…”

Section: Introductionmentioning

confidence: 99%

“…Clearly, its image under restriction is nothing but the ([K : F]dimensional) left Bernoulli shift of Res K F (V ) = ∞ n=0 F [K:F] . Therefore, ent * F (Res K F (V, K β)) = ent * F ( ∞ n=0 [K:F] , F [K:F] β) = [K : F] = [K : F] · ent * K (V, K β),by[5, Example 3.13(b)]. On the other hand, the induction gives rise to the (1-dimensional) left Bernoulli shift on Ind L K…”

mentioning

confidence: 98%

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Topological entropy for locally linearly compact vector spaces and field extensions

Castellano

2020

Topological Algebra and Its Applications

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Section: Topological Entropy On K Llcmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 98%

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Topological entropy for locally linearly compact vector spaces and field extensions

Castellano

2020

Topological Algebra and Its Applications

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“…In particular, the algebraic dimension entropy ent dim for discrete vector spaces was thoroughly investigated in [61], and carried to locally linearly compact vector spaces in [21]. Also a topological dimension entropy ent ⋆ dim was studied in [22] for locally linearly compact vector spaces.…”

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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Algebraic Entropy in Locally Linearly Compact Vector Spaces

Castellano

Bruno

2017

Rings, Polynomials, and Modules

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We introduce the algebraic entropy for continuous endomorphisms of locally linearly compact vector spaces over a discrete field, as the natural extension of the algebraic entropy for endomorphisms of discrete vector spaces studied in [10]. We show that the main properties of entropy continue to hold in the general context of locally linearly compact vector spaces, in particular we extend the Addition Theorem.

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Topological entropy for locally linearly compact vector spaces

Cited by 11 publications

References 37 publications

Topological entropy for locally linearly compact vector spaces and field extensions

Topological entropy for locally linearly compact vector spaces and field extensions

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

Algebraic Entropy in Locally Linearly Compact Vector Spaces

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