Algebraic Entropy in Locally Linearly Compact Vector Spaces

Castellano, Ilaria; Bruno, Anna Giordano

doi:10.1007/978-3-319-65874-2_6

Cited by 11 publications

(23 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The same occurs for the algebraic and the topological dimension entropy for locally linearly compact vector spaces from [21,22], indeed in principle their are as the intrinsic entropy.…”

Section: Final Remarksmentioning

confidence: 72%

“…The adjoint version of the algebraic i-entropy was investigated in [106]. 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%

“…Remark 5.37. In [21] and [22], the algebraic and the topological dimension entropy are studied in the more general case of locally linearly compact vector spaces (a linearly topologized vector space V over a discrete field K is locally linearly compact if it admits a local base at 0 consisting of linearly compact open linear subspaces). As it occurs for the intrinsic algebraic entropy, these entropies do not fit our general scheme.…”

Section: Algebraic and Topological Entropy For Vector Spacesmentioning

confidence: 99%

See 2 more Smart Citations

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

“…The same occurs for the algebraic and the topological dimension entropy for locally linearly compact vector spaces from [21,22], indeed in principle their are as the intrinsic entropy.…”

Section: Final Remarksmentioning

confidence: 72%

Section: Historical Backgroundmentioning

confidence: 99%

Section: Algebraic and Topological Entropy For Vector Spacesmentioning

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

show abstract

“…Therefore, going down the same path, one defines the intrinsic dimension entropy ent dim for linear endomorphisms by In this new context, Corollary 2 can be then used to compute the intrinsic dimension entropy of the two sided Bernoulli shift β K , which turns out to equal 1. Indeed, Corollary 2 and a limit-free formula as in [4,15] provide ent dim (β K ) = dim K (V − /β −1 K (V − )) = 1. Quite remarkably, φ-inert subspaces do not enrich the dynamics of linear flows like φ-inert subgroups do in the framework of abelian groups (see [9]).…”

Section: Connection With Algebraic Entropymentioning

confidence: 95%

“…It was introduced in [9] to obtain a dynamical invariant able to treat also endomorphisms of torsion-free abelian groups where other entropy functions vanish completely for the lack of non-trivial finite subgroups. Afterwards, the intrinsic valuation entropy ent v was introduced in [15] with the aim of extending ent to the context of modules over a non-discrete valuation domain and also the algebraic entropy for locally linearly compact vector spaces defined in [4] has the same "intrinsic" flavour. Therefore, going down the same path, one defines the intrinsic dimension entropy ent dim for linear endomorphisms by In this new context, Corollary 2 can be then used to compute the intrinsic dimension entropy of the two sided Bernoulli shift β K , which turns out to equal 1.…”

Section: Connection With Algebraic Entropymentioning

confidence: 99%

A property of the lamplighter group

Castellano

Cook

Kropholler

2020

Topological Algebra and Its Applications

View full text Add to dashboard Cite

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.

show abstract

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

Castellano

2020

Journal of Pure and Applied Algebra

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Algebraic Entropy in Locally Linearly Compact Vector Spaces

Cited by 11 publications

References 28 publications

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

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

A property of the lamplighter group

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

Contact Info

Product

Resources

About