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A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Geometry and Topology".

Deadline for manuscript submissions: closed (10 November 2022) | Viewed by 10131
Please contact the Guest Editor or the Journal Editor (alex.zhang@mdpi.com) for any queries about the scope, discount, submission procedure and publication process.

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Special Issue Editor

Prof. Dr. Salvador Hernández

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Guest Editor

Departament de Matemàtiques, Universitat Jaume I, 12071 Castellón de la Plana, Spain
Interests: topological groups; general topology; functional analysis

Special Issue Information

Dear Colleagues,

The origin of topological groups dates back to Lie, Poincaré, Hilbert, Brower and Screier, who coined the term “Abstrakte kontinuierliche Gruppen”. Since then, the theory of topological groups has become a crossroads in several important areas of mathematics: algebra, topology, functional and harmonic analysis and topological dynamics, to name a few.

This Special Issue aims to collect high-quality research papers concerning all research lines where topological groups and the relation to topological dynamics (that is, dynamical systems of the form (G,X), where X is a topological space and the group G acts by homeomorphisms) have a relevant role. We welcome contributions that deal with recent developments in these areas.

Topics for this Special Issue include topological groups, Lie groups and Lie algebras, discrete subgroups of Lie groups, locally compact abelian groups (LCA groups), compact groups, locally compact groups and their algebras, transformation groups and semigroups, Lie transformation groups and holomorphic transformation groups, abstract harmonic analysis, quantum groups and enveloping algebras, Kac–Moody groups and algebras, topological and differentiable algebraic systems, topological dynamics and dynamical systems over complex numbers.

Prof. Dr. Salvador Hernández
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Axioms is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

topological groups
compact groups
profinite groups
prodiscrete groups
locally compact groups
lie groups
pro-lie groups
semitopological groups
representations
group algebras
compactifications of topological groups and semigroups
free topological groups and free products
variety of topological groups
duality and reflexivity
covering theory for topological groups
algebraic topology and topological groups
transformation groups and semigroups
groups of homeomorphisms or diffeomorphisms
group actions
homogeneous spaces
positive definite functions on groups, semigroups
character groups and dual objects
spectral synthesis on groups, semigroups, etc.
special sets (thin sets, kronecker sets, helsonsets, ditkin sets, sidon sets, etc.)
almost periodic functions on groups and semigroups

Published Papers (6 papers)

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Research

11 pages, 296 KiB

Open AccessArticle

Orbit Tracing Properties on Hyperspaces and Fuzzy Dynamical Systems

by Salud Bartoll, Félix Martínez-Giménez, Alfred Peris and Francisco Rodenas

Axioms 2022, 11(12), 733; https://doi.org/10.3390/axioms11120733 - 15 Dec 2022

Viewed by 1452

Abstract

Let X be a compact metric space and a continuous map

f : X \to X

which defines a discrete dynamical system

(X, f)

. The map f induces two natural maps, namely [...] Read more.

Let X be a compact metric space and a continuous map

f : X \to X

which defines a discrete dynamical system

(X, f)

. The map f induces two natural maps, namely

\bar{f} : K (X) \to K (X)

on the hyperspace

K (X)

of non-empty compact subspaces of X and the Zadeh’s extension

\hat{f} : F (X) \to F (X)

on the space

F (X)

of normal fuzzy set. In this work, we analyze the interaction of some orbit tracing dynamical properties, namely the specification and shadowing properties of the discrete dynamical system

(X, f)

and its induced discrete dynamical systems

(K (X), \bar{f})

and

(F (X), \hat{f})

. Adding an algebraic structure yields stronger conclusions, and we obtain a full characterization of the specification property in the hyperspace, in the fuzzy space, and in the phase space X if we assume that the later is a convex compact subset of a (metrizable and complete) locally convex space and f is a linear operator. Full article

(This article belongs to the Special Issue Topological Groups and Dynamics)

8 pages, 276 KiB

Open AccessArticle

Complete Invariant ⋆-Metrics on Semigroups and Groups

by Shi-Yao He, Jian-Cai Wei and Li-Hong Xie

Axioms 2022, 11(10), 546; https://doi.org/10.3390/axioms11100546 - 11 Oct 2022

Cited by 1 | Viewed by 1069

Abstract

In this paper, we study the complete ⋆-metric semigroups and groups and the Raǐkov completion of invariant ⋆-metric groups. We obtain the following. (1) Let

(X, d^{⋆})

be a complete ⋆-metric space containing a semigroup (group) G that is [...] Read more.

In this paper, we study the complete ⋆-metric semigroups and groups and the Raǐkov completion of invariant ⋆-metric groups. We obtain the following. (1) Let

(X, d^{⋆})

be a complete ⋆-metric space containing a semigroup (group) G that is a dense subset of X. If the restriction of

d^{⋆}

on G is invariant, then X can become a semigroup (group) containing G as a subgroup, and

d^{⋆}

is invariant on X. (2) Let

(G, d^{⋆})

be a ⋆-metric group such that

d^{⋆}

is invariant on G. Then,

(G, d^{⋆})

is complete if and only if

(G, τ_{d^{⋆}})

is Raǐkov complete. Full article

(This article belongs to the Special Issue Topological Groups and Dynamics)

13 pages, 388 KiB

Open AccessArticle

A Selection Principle and Products in Topological Groups

by Marion Scheepers

Axioms 2022, 11(6), 286; https://doi.org/10.3390/axioms11060286 - 13 Jun 2022

Cited by 1 | Viewed by 1477

Abstract

We consider the preservation under products, finite powers, and forcing of a selection-principle-based covering property of

T_{0}

topological groups. Though the paper is partly a survey, it contributes some new information: (1) The product of a strictly o-bounded group with an o-bounded [...] Read more.

We consider the preservation under products, finite powers, and forcing of a selection-principle-based covering property of

T_{0}

topological groups. Though the paper is partly a survey, it contributes some new information: (1) The product of a strictly o-bounded group with an o-bounded group is an o-bounded group—Corollary 1. (2) In the generic extension by a finite support iteration of

ℵ_{1}

Hechler reals the product of any o-bounded group with a ground model

ℵ_{0}

bounded group is an o-bounded group—Theorem 11. (3) In the generic extension by a countable support iteration of Mathias reals the product of any o-bounded group with a ground model

ℵ_{0}

bounded group is an o-bounded group—Theorem 12. Full article

(This article belongs to the Special Issue Topological Groups and Dynamics)

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Figure 1

17 pages, 321 KiB

Open AccessArticle

Macdonald Formula, Ricci Curvature, and Concentration Locus for Classical Compact Lie Groups

by Sergio Cacciatori and Pietro Ursino

Axioms 2022, 11(6), 245; https://doi.org/10.3390/axioms11060245 - 24 May 2022

Cited by 2 | Viewed by 1249

Abstract

We consider the phenomenon of concentration of measures, which is restricted to the case of families of compact connected Lie groups. While in the literature, powerful general results regarding the existence of concentration and its relations to extremal amenability of infinite dimensional groups have been determined, there are few explicit examples, specially regarding the determination of the region where the measure concentrates. Since they can be relevant for concrete applications, both in mathematics and in physics, in the present paper, we provide a number of such examples, using compact Lie groups as basic ingredients. In particular, our strategy is to employ the Macdonald’s formula, giving the volume of compact simple Lie groups, and Ricci curvature of the bi-invariant metric for analyzing a “concentration locus”, which is a tool to detect where a sequence of metric, Borel measurable spaces concentrates its measure. Full article

(This article belongs to the Special Issue Topological Groups and Dynamics)

10 pages, 274 KiB

Open AccessFeature PaperArticle

Homomorphic Encoders of Profinite Abelian Groups II

by María V. Ferrer and Salvador Hernández

Axioms 2022, 11(4), 158; https://doi.org/10.3390/axioms11040158 - 29 Mar 2022

Cited by 1 | Viewed by 1421

Abstract

Let

{G_{i} : i \in N}

be a family of finite Abelian groups. We say that a subgroup

G \leq \prod_{i \in N} G_{i}

is order controllable if for every

i \in N

, there is [...] Read more.

Let

{G_{i} : i \in N}

be a family of finite Abelian groups. We say that a subgroup

G \leq \prod_{i \in N} G_{i}

is order controllable if for every

i \in N

, there is

n_{i} \in N

such that for each

c \in G

, there exists

c_{1} \in G

satisfying

c_{1 | [1, i]} = c_{| [1, i]}

s u p p (c_{1}) \subseteq [1, n_{i}]

, and order

(c_{1})

divides order

(c_{| [1, n_{i}]})

. In this paper, we investigate the structure of order-controllable group codes. It is proved that if G is an order controllable, shift invariant, group code over a finite abelian group H, then G possesses a finite canonical generating set. Furthermore, our construction also yields that G is algebraically conjugate to a full group shift. Full article

(This article belongs to the Special Issue Topological Groups and Dynamics)

6 pages, 268 KiB

Open AccessArticle

Products of K-Analytic Sets in Locally Compact Groups and Kuczma–Ger Classes

by Taras Banakh, Iryna Banakh and Eliza Jabłońska

Axioms 2022, 11(2), 65; https://doi.org/10.3390/axioms11020065 - 07 Feb 2022

Viewed by 1568

Abstract

We prove that for any K-analytic subsets

A, B

of a locally compact group X if the product

A B

has empty interior (and is meager) in X, then one of the sets A or B can be covered by [...] Read more.

We prove that for any K-analytic subsets

A, B

of a locally compact group X if the product

A B

has empty interior (and is meager) in X, then one of the sets A or B can be covered by countably many closed nowhere dense subsets (of Haar measure zero) in X. This implies that a K-analytic subset A of X can be covered by countably many closed Haar-null sets if the set

A A A A

has an empty interior in X. It also implies that every non-open K-analytic subgroup of a locally compact group X can be covered by countably many closed Haar-null sets in X (for analytic subgroups of the real line this fact was proved by Laczkovich in 1998). Applying this result to the Kuczma–Ger classes, we prove that an additive function

f : X \to R

on a locally compact topological group X is continuous if and only if f is upper bounded on some K-analytic subset

A \subseteq X

that cannot be covered by countably many closed Haar-null sets. Full article

(This article belongs to the Special Issue Topological Groups and Dynamics)

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