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Dikran Dikranjan

Publications and source records attributed to Dikran Dikranjan.

At least 19 recordsLinked to original sources
arXiv2025

The Protasov-Zelenyuk topology and ideal convergence

The so-called $T$-sequences $\mathbf u$ in a group $G$, and the related finest Hausdorff group topology $T_\mathbf u$ on $G$ that makes $\mathbf u$ a null sequence, were introduced by Protasov and Zelenyuk 35 years ago and since then they became a fundamental tool in the field of topological groups. More recently, in the abelian case, the subfamily of $T$-sequences called $TB$-sequences was introduced, as well as the finest precompact group topology $T_\mathbf{pu}$ that makes $\mathbf u$ a null sequence. Here we study the counterpart of all these notions with respect to ideal convergence in place of the classical notion of convergence of a sequence. Also, we study their relation to the already established field of $I$-characterized subgroups of compact abelian groups.

math.GR↗
arXiv2025

The Structure of Sequentially Complete Locally Minimal Groups

Generalizing results from \cite{DTk,DU} we study the fine structure of locally minimal (locally) precompact Abelian groups (these are the locally essential subgroups $G$ of LCA groups $L$, i.e., such that $G$ non-trivially meets all ``small" closed subgroup of $L$). More precisely we prove that if $G$ is a dense locally minimal and sequentially closed subgroup of a LCA group $L$, then the connected component $c(G)$ of $G$ has the same weight as $c(L)$. Moreover, when $w(c(G))$ is not Ulam measurable, then $c(G) = c(L)$. We provide an extended discussion illustrating how this result fails in various ways in the non-abelian case (even for nilpotent groups of class 2). Motivated by the above result, we study further those locally minimal precompact Abelian groups $G$, termed {\em critical locally minimal},such that $c(G) =c(K)$ (where $K$ is the compact completion of $G$) and $G/c(G)$ is not locally minimal. Such a group cannot be compact, neither connected, nor totally disconnected. We provide a proper class of critical locally minimal groups with additional compactness-like properties and we study the class $\CCC$ of compact Abelian groups with a dense critical locally minimal subgroup. In particular, we completely describe the connected components of the finite-dimensional groups belonging to $\CCC$.

math.GR↗
arXiv2025

Element-wise description of the $\mathcal I$-characterized subgroups of the circle

According to Cartan, given an ideal $\mathcal I$ of $\mathbb N$, a sequence $(x_n)_{n\in\mathbb N}$ in the circle group $\mathbb T$ is said to {\em $\mathcal I$-converge} to a point $x\in \mathbb T$ if $\{n\in \mathbb N: x_n \not \in U\}\in \mathcal I$ for every neighborhood $U$ of $x$ in $\mathbb T$. For a sequence $\mathbf u=(u_n)_{n\in\mathbb N}$ in $\mathbb Z$, let $$t_\mathbf u^\mathcal I(\mathbb T) :=\{x\in \mathbb T: u_nx \ \text{$\mathcal I$-converges to}\ 0 \}.$$ For an analytic free $P$-ideal $\mathcal I$, this set is a Borel (hence, Polishable) subgroup of $\mathbb T$ with many nice properties, largely studied in the case when $\mathcal I = \mathcal F in$ is the ideal of all finite subsets of $\mathbb N$ (so $\mathcal F in$-convergence coincides with the usual one) for its remarkable connection to topological algebra, descriptive set theory and harmonic analysis. We give a complete element-wise description of $t_\mathbf u^\mathcal I(\mathbb T)$ when $\mathbf u$ is a strictly increasing sequence of positive integers with $u_0=1$ and $u_n\mid u_{n+1}$ for every $n\in\mathbb N$ and under suitable hypotheses on $\mathcal I$. In the special case when $\mathcal I =\mathcal F in$, we obtain an alternative proof of a simplified version of a known result.

math.GN↗
arXiv2023

The algebraic entropy of one-dimensional finitary linear cellular automata

The aim of this paper is to present one-dimensional finitary linear cellular automata $S$ on $\mathbb Z_m$ from an algebraic point of view. Among various other results, we: (i) show that the Pontryagin dual $\widehat S$ of $S$ is a classical one-dimensional linear cellular automaton $T$ on $\mathbb Z_m$; (ii) give several equivalent conditions for $S$ to be invertible with inverse a finitary linear cellular automaton; (iii) compute the algebraic entropy of $S$, which coincides with the topological entropy of $T=\widehat S$ by the so-called Bridge Theorem. In order to better understand and describe the entropy we introduce the degree $\mathrm{deg}(S)$ and $\mathrm{deg}(T)$ of $S$ and $T$.

math.GR↗
arXiv2023

Ore localization of amenable monoid actions and applications towards entropy $-$ addition formulas and the bridge theorem

For a left action $S\oversetλ{\curvearrowright}X$ of a cancellative right amenable monoid $S$ on a discrete Abelian group $X$, we construct its Ore localization $G\overset{λ^*}{\curvearrowright}X^*$, where $G$ is the group of left fractions of $S$; analogously, for a right action $K\oversetρ\curvearrowleft S$ on a compact space $K$, we construct its Ore colocalization $K^*\overset{ρ^*}{\curvearrowleft} G$. Both constructions preserve entropy, i.e., for the algebraic entropy $h_{\mathrm{alg}}$ and for the topological entropy $h_{\mathrm{top}}$ one has $h_{\mathrm{alg}}(λ)=h_{\mathrm{alg}}(λ^*)$ and $h_{\mathrm{top}}(ρ)=h_{\mathrm{top}}(ρ^*)$, respectively. Exploiting these constructions and the theory of quasi-tilings, we extend the Addition Theorem for $h_{\mathrm{top}}$, known for right actions of countable amenable groups on compact metrizable groups, to right actions $K\oversetρ{\curvearrowleft} S$ of cancellative right amenable monoids $S$ (with no restrictions on the cardinality) on arbitrary compact groups $K$. When the compact group $K$ is Abelian, we prove that $h_{\mathrm{top}}(ρ)$ coincides with $h_{\mathrm{alg}}(\hatρ)$, where $S\overset{\hatρ}\curvearrowright X$ is the dual left action on the discrete Pontryagin dual $X=\hat{K}$, that is, a so-called Bridge Theorem. From the Addition Theorem for $h_{\mathrm{top}}$ and the Bridge Theorem, we obtain an Addition Theorem for $h_{\mathrm{alg}}$ for left actions $S\oversetλ\curvearrowright X$ on discrete Abelian groups, so far known only under the hypotheses that either $X$ is torsion or $S$ is locally monotileable. The proofs substantially use the unified approach towards entropy based on the entropy of actions of cancellative right amenable monoids on appropriately defined normed monoids.

math.RT↗
arXiv2021

Countably compact groups having minimal infinite powers

We answer the question, raised more than thirty years ago, on whether the power (G raised to the power omega) of a countably compact minimal Abelian group G is minimal, by showing that the negative answer is equivalent to the existence of measurable cardinals.

math.LO↗
arXiv2021

Groups with cofinite Zariski topology and potential density

Tkachenko and Yaschenko [34] characterized the abelian groups G such that all proper unconditionally closed subsets of G are finite, these are precisely the abelian groups G having cofinite Zariski topology (they proved that such a G is either almost torsion-free or of prime exponent). The authors connected this fact to Markov's notion of potential density and the existence of pairs of independent group topologies. Inspired by their work, we examine the class C of groups having cofinite Zariski topology in the general case, obtaining a number of very strong restrictions on these groups in the non-abelian case which suggest the bold conjecture that a group with cofinite Zariski topology is necessarily either abelian or finite. We show that Tkachenko-Yaschenko theorem fails in the non-abelian case and we offer a natural counterpart in the general case using a partial Zariski topology and an appropriate stronger version of the property almost torsion-free.

math.GR↗
arXiv2021

Metric vs topological receptive entropy of semigroup actions

We study the receptive metric entropy for semigroup actions on probability spaces, inspired by a similar notion of topological entropy introduced by Hofmann and Stoyanov. We analyze its basic properties and its relation with the classical metric entropy. In the case of semigroup actions on compact metric spaces we compare the receptive metric entropy with the receptive topological entropy looking for a Variational Principle. With this aim we propose several characterizations of the receptive topological entropy. Finally we introduce a receptive local metric entropy inspired by a notion by Bowen generalized in the classical setting of amenable group actions by Zheng and Chen, and we prove partial versions of the Brin-Katok Formula and the local Variational Principle.

math.DS↗
arXiv2020

Intrinsic entropy for generalized quasimetric semilattices

We introduce the notion of intrinsic semilattice entropy $\widetilde h$ in the category $\mathcal L_{qm}$ of generalized quasimetric semilattices and contractive homomorphisms. By using appropriate categories $\mathfrak X$ and functors $F:\mathfrak X\to\mathcal L_{qm}$ we find specific known entropies $\widetilde h_\mathfrak X$ on $\mathfrak X$ as intrinsic functorial entropies, that is, as $\widetilde h_\mathfrak X=\widetilde h\circ F$. These entropies are the intrinsic algebraic entropy, the algebraic and the topological entropies for locally linearly compact vector spaces, the topological entropy for locally compact totally disconnected groups and the algebraic entropy for locally compact compactly covered abelian groups.

math.GR↗
arXiv2020

Finiteness of topological entropy for locally compact abelian groups

We study the locally compact abelian groups in the class $\mathfrak E_{<\infty}$, that is, having only continuous endomorphisms of finite topological entropy, and in its subclass $\mathfrak E_0$, 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 $\mathfrak E_{<\infty}$, and that those of them that belong to $\mathfrak E_0$ 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.

math.DS↗
arXiv2020

The addition theorem for locally monotileable monoid actions

We prove an instance of the so-called Addition Theorem for the algebraic entropy of actions of cancellative right amenable monoids $S$ on discrete abelian groups $A$ by endomorphisms, under the hypothesis that $S$ is locally monotileable (that is, $S$ admits a right Følner sequence $(F_n)_{n\in\mathbb N}$ such that $F_n$ is a monotile of $F_{n+1}$ for every $n\in\mathbb N$). We study in details the class of locally monotileable groups, also in relation with already existing notions of monotileability for groups, introduced by Weiss and developed further by other authors recently.

math.GR↗
arXiv2019

Cardinal invariants and convergence properties of locally minimal groups

If G is a locally essential subgroup of a compact abelian group K, then: (i) t(G)=w(G)=w(K), where t(G) is the tightness of G; (ii) if G is radial, then K must be metrizable; (iii) G contains a super-sequence S converging to 0 such that |S|=w(G)=w(K). Items (i)--(iii) hold when G is a dense locally minimal subgroup of K. We show that locally minimal, locally precompact abelian groups of countable tightness are metrizable. In particular, a minimal abelian group of countable tightness is metrizable. This answers a question of O. Okunev posed in 2007. For every uncountable cardinal kappa, we construct a Frechet-Urysohn minimal group G of character kappa such that the connected component of G is an open normal omega-bounded subgroup (thus, G is locally precompact). We also build a minimal nilpotent group of nilpotency class 2 without non-trivial convergent sequences having an open normal countably compact subgroup.

math.GN↗
arXiv2019

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

We present a unifying approach to the study of entropies in Mathematics, such as measure entropy, topological entropy, algebraic entropy, set-theoretic entropy. We take into account discrete dynamical systems, that is, pairs $(X,T)$, where $X$ is the underlying space and $T:X\to X$ a transformation. We see entropies as functions $h:\mathfrak X\to \mathbb R_+$, associating to each flow $(X,T)$ of a category $\mathfrak X$ either a non negative real or $\infty$. We introduce the notion of semigroup entropy $h_\mathfrak S:\mathfrak S\to\mathbb R_+$, which is a numerical invariant attached to endomorphisms of the category $\mathfrak S$ of normed semigroups. Then, for a functor $F:\mathfrak X\to\mathfrak S$ from any specific category $\mathfrak X$ to $\mathfrak S$, we define the functorial entropy $h_F:\mathfrak X\to\mathbb R_+$ as the composition $h_{\mathfrak S}\circ F$. Clearly, $h_F$ inherits many of the properties of $h_\mathfrak S$, depending also on the properties of $F$. Such general scheme permits to obtain relevant known entropies as functorial entropies $h_F$, for appropriate categories $\mathfrak X$ and functors $F$, and to establish the properties shared by them. 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 formalize the notion of Bridge Theorem between two entropies $h_i:\mathfrak X_i\to \mathbb R_+$, $i=1,2$, with respect to a functor $\varepsilon:\mathfrak X_1\to\mathfrak X_2$. Then, for pairs of functorial entropies we use the above scheme to introduce the notion and the related scheme of Strong Bridge Theorem, which allows us to put under the same umbrella various relations between pairs of entropies.

math.DS↗
arXiv2019

Algebraic entropy for amenable semigroup actions

We introduce two notions of algebraic entropy for actions of cancellative right amenable semigroups $S$ on discrete abelian groups $A$ by endomorphisms; these extend the classical algebraic entropy for endomorphisms of abelian groups, corresponding to the case $S=\mathbb N$. We investigate the fundamental properties of the algebraic entropy and compute it in several examples, paying special attention to the case when S is an amenable group. For actions of cancellative right amenable monoids on torsion abelian groups, we prove the so called Addition Theorem. In the same setting, we see that a Bridge Theorem connects the algebraic entropy with the topological entropy of the dual action by means of the Pontryagin duality, so that we derive an Addition Theorem for the topological entropy of actions of cancellative left amenable monoids on totally disconnected compact abelian groups.

math.GR↗
arXiv2019

Categories of coarse groups: quasi-homomorphisms and functorial coarse structures

Coarse geometry is the study of large-scale properties of spaces. In this paper we study group coarse structures (i.e., coarse structures on groups that agree with the algebraic structures), by using group ideals. We introduce a large class of examples of group coarse structures induced by cardinal invariants. In order to enhance the categorical treatment of the subject, we use quasi-homomorphisms, as a large-scale counterpart of homomorphisms. In particular, the localisation of a category plays a fundamental role. We then define the notion of functorial coarse structures and we give various examples of those structures.

math.GN↗
arXiv2018

Densely locally minimal groups

We study locally compact groups having all dense subgroups (locally) minimal. We call such groups densely (locally) minimal. In 1972 Prodanov proved that the infinite compact abelian groups having all subgroups minimal are precisely the groups $\mathbb Z_p$ of $p$-adic integers. In [31], we extended Prodanov's theorem to the non-abelian case at several levels. In this paper, we focus on the densely (locally) minimal abelian groups. We prove that in case that a topological abelian group $G$ is either compact or connected locally compact, then $G$ is densely locally minimal if and only if $G$ either is a Lie group or has an open subgroup isomorphic to $\mathbb Z_p$ for some prime $p$. This should be compared with the main result of [9]. Our Theorem C provides another extension of Prodanov's theorem: an infinite locally compact group is densely minimal if and only if it is isomorphic to $\mathbb Z_p$. In contrast, we show that there exists a densely minimal, compact, two-step nilpotent group that neither is a Lie group nor it has an open subgroup isomorphic to $\mathbb Z_p$.

math.GN↗
arXiv2018

Hereditarily minimal topological groups

We study locally compact groups having all subgroups minimal. We call such groups hereditarily minimal. In 1972 Prodanov proved that the infinite hereditarily minimal compact abelian groups are precisely the groups $\mathbb Z_p$ of $p$-adic integers. We extend Prodanov's theorem to the non-abelian case at several levels. For infinite hypercentral (in particular, nilpotent) locally compact groups we show that the hereditarily minimal ones remain the same as in the abelian case. On the other hand, we classify completely the locally compact solvable hereditarily minimal groups, showing that in particular they are always compact and metabelian. The proofs involve the (hereditarily) locally minimal groups, introduced similarly. In particular, we prove a conjecture by He, Xiao and the first two authors, showing that the group $\mathbb Q_p\rtimes \mathbb Q_p^*$ is hereditarily locally minimal, where $\mathbb Q_p^*$ is the multiplicative group of non-zero $p$-adic numbers acting on the first component by multiplication. Furthermore, it turns out that the locally compact solvable hereditarily minimal groups are closely related to this group.

math.GN↗
arXiv2017

Generalize Heisenberg Groups and Self-Duality

This paper compares two generalizations of Heisenberg groups and studies their connection to one of the major open problems in the field of locally compact abelian groups, namely the description of the self-dual locally compact abelian groups ([12]). The first generalization is presented by the so called generalized Heisenberg groups $\mathbb{H}(ω)$, defined in analogy with the classical Heisenberg group and the second one is inspired by the construction proposed by Mumford in [16] and named after him as Weyl-Mumford groups (WM groups). These two families can be defined also in the framework of topological groups. We investigate the relationship between locally compact WM groups, locally compact Generalized Heisenberg with center isomorphic to $\mathbb{T}$ and symplectic self-dualities.

math.GR↗