scholarly journals Embeddings into monothetic groups

2018 ◽  
Vol 21 (4) ◽  
pp. 579-581
Author(s):  
Michal Doucha
Keyword(s):  
Topological Group ◽  
Abelian Group ◽  
Elementary Proof ◽  
Invariant Metric ◽  
Abelian Topological Group

AbstractWe provide a very short elementary proof that every separable abelian group with a bounded invariant metric isometrically embeds into a monothetic group with a bounded invariant metric, in such a way that the result of Morris and Pestov that every separable abelian topological group embeds into a monothetic group is an immediate corollary. We show that the boundedness assumption cannot be dropped.

scholarly journals Convergence of ergodic averages for many group rotations

2015 ◽  
Vol 36 (7) ◽  
pp. 2107-2120
Author(s):  
ZOLTÁN BUCZOLICH ◽  
GABRIELLA KESZTHELYI
Keyword(s):  
Topological Group ◽  
Abelian Group ◽  
Measurable Function ◽  
Dual Group ◽  
Ergodic Averages ◽  
Abelian Topological Group ◽  
Compact Abelian Topological Group ◽  
Image Position ◽  
Rotation Set ◽  
Group Rotations

Suppose that $G$ is a compact Abelian topological group, $m$ is the Haar measure on $G$ and $f:G\rightarrow \mathbb{R}$ is a measurable function. Given $(n_{k})$, a strictly monotone increasing sequence of integers, we consider the non-conventional ergodic/Birkhoff averages $$\begin{eqnarray}M_{N}^{\unicode[STIX]{x1D6FC}}f(x)=\frac{1}{N+1}\mathop{\sum }_{k=0}^{N}f(x+n_{k}\unicode[STIX]{x1D6FC}).\end{eqnarray}$$ The $f$-rotation set is $$\begin{eqnarray}\unicode[STIX]{x1D6E4}_{f}=\{\unicode[STIX]{x1D6FC}\in G:M_{N}^{\unicode[STIX]{x1D6FC}}f(x)\text{ converges for }m\text{ almost every }x\text{ as }N\rightarrow \infty \}.\end{eqnarray}$$We prove that if $G$ is a compact locally connected Abelian group and $f:G\rightarrow \mathbb{R}$ is a measurable function then from $m(\unicode[STIX]{x1D6E4}_{f})>0$ it follows that $f\in L^{1}(G)$. A similar result is established for ordinary Birkhoff averages if $G=Z_{p}$, the group of $p$-adic integers. However, if the dual group, $\widehat{G}$, contains ‘infinitely many multiple torsion’ then such results do not hold if one considers non-conventional Birkhoff averages along ergodic sequences. What really matters in our results is the boundedness of the tail, $f(x+n_{k}\unicode[STIX]{x1D6FC})/k$, $k=1,\ldots ,$ for almost every $x$ for many $\unicode[STIX]{x1D6FC}$; hence, some of our theorems are stated by using instead of $\unicode[STIX]{x1D6E4}_{f}$ slightly larger sets, denoted by $\unicode[STIX]{x1D6E4}_{f,b}$.

scholarly journals Groups – Additive Notation

2015 ◽  
Vol 23 (2) ◽  
pp. 127-160 ◽  
Author(s):  
Roland Coghetto
Keyword(s):  
Topological Group ◽  
Group Theory ◽  
Abelian Group ◽  
Normal Subgroup ◽  
Dense Subset ◽  
Abelian Topological Group ◽  
Covering Group ◽  
Mizar Mathematical Library ◽  
Order Of An Element ◽  
Generic Theory

Abstract We translate the articles covering group theory already available in the Mizar Mathematical Library from multiplicative into additive notation. We adapt the works of Wojciech A. Trybulec [41, 42, 43] and Artur Korniłowicz [25]. In particular, these authors have defined the notions of group, abelian group, power of an element of a group, order of a group and order of an element, subgroup, coset of a subgroup, index of a subgroup, conjugation, normal subgroup, topological group, dense subset and basis of a topological group. Lagrange’s theorem and some other theorems concerning these notions [9, 24, 22] are presented. Note that “The term ℤ-module is simply another name for an additive abelian group” [27]. We take an approach different than that used by Futa et al. [21] to use in a future article the results obtained by Artur Korniłowicz [25]. Indeed, Hölzl et al. showed that it was possible to build “a generic theory of limits based on filters” in Isabelle/HOL [23, 10]. Our goal is to define the convergence of a sequence and the convergence of a series in an abelian topological group [11] using the notion of filters.

Homeomorphism and Isomorphism of Abelian Groups

1974 ◽  
Vol 26 (6) ◽  
pp. 1515-1519 ◽  
Author(s):  
Stephen Scheinberg
Keyword(s):  
Topological Group ◽  
Abelian Group ◽  
Topological Space ◽  
Abelian Groups ◽  
Group Structure ◽  
Locally Compact ◽  
Abelian Topological Group

An abelian topological group can be considered simply as an abelian group or as a topological space. The question considered in this article is whether the topological group structure is determined by these weaker structures. Denote homeomorphism, isomorphism, and homeomorphic isomorphism by ≈, ≅ , and =, respectively. The principal results are these.Theorem 1. If G1andG2are locally compact and connected, then G1≈ G2implies G1= G2.

scholarly journals CONTINUITY OF MEASURABLE HOMOMORPHISMS

2008 ◽  
Vol 78 (1) ◽  
pp. 171-176 ◽  
Author(s):  
JANUSZ BRZDȨK
Keyword(s):  
Topological Group ◽  
Abelian Group ◽  
Baire Property ◽  
Topological Groups ◽  
Locally Compact ◽  
Compact Topological Group ◽  
Locally Compact Topological Group

AbstractWe give some general results concerning continuity of measurable homomorphisms of topological groups. As a consequence we show that a Christensen measurable homomorphism of a Polish abelian group into a locally compact topological group is continuous. We also obtain similar results for the universally measurable homomorphisms and the homomorphisms that have the Baire property.

scholarly journals A Lebesgue decomposition for elements in a topological group

1980 ◽  
Vol 3 (4) ◽  
pp. 801-808
Author(s):  
Thomas P. Dence
Keyword(s):  
Topological Group ◽  
Abelian Group ◽  
Projection Operators ◽  
Additive Functions ◽  
Lebesgue Decomposition ◽  
Algebra Of Sets

Our aim is to establish the Lebesgue decomposition for strongly-bounded elements in a topological group. In 1963 Richard Darst established a result giving the Lebesgue decomposition of strongly-bounded elements in a normed Abelian group with respect to an algebra of projection operators. Consequently, one can establish the decomposition of strongly-bounded additive functions defined on an algebra of sets. Analagous results follow for lattices of sets. Generalizing some of the techniques yield decompositions for elements in a topological group.

The Meet Operator in the Lattice of Group Topologies

1986 ◽  
Vol 29 (4) ◽  
pp. 478-481
Author(s):  
Bradd Clark ◽  
Victor Schneider
Keyword(s):  
Topological Group ◽  
Abelian Group ◽  
Locally Compact Group ◽  
Group Topology ◽  
Locally Compact ◽  
Hausdorff Topology ◽  
Lattice Of Topologies ◽  
Group Form ◽  
Complemented Lattice ◽  
Infinite Abelian Group

AbstractIt is well known that the lattice of topologies on a set forms a complete complemented lattice. The set of topologies which make G into a topological group form a complete lattice L(G) which is not a sublattice of the lattice of all topologies on G.Let G be an infinite abelian group. No nontrivial Hausdorff topology in L(G) has a complement in L(G). If τ1 and τ2 are locally compact topologies then τ1Λτ2 is also a locally compact group topology. The situation when G is nonabelian is also considered.

scholarly journals VARIETIES OF ABELIAN TOPOLOGICAL GROUPS AND SCATTERED SPACES

2008 ◽  
Vol 78 (3) ◽  
pp. 487-495 ◽  
Author(s):  
CAROLYN E. MCPHAIL ◽  
SIDNEY A. MORRIS
Keyword(s):  
Topological Group ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Topological Groups ◽  
Abelian Topological Group ◽  
Free Abelian Topological Group ◽  
Scattered Spaces

AbstractThe variety of topological groups generated by the class of all abelian kω-groups has been shown to equal the variety of topological groups generated by the free abelian topological group on [0, 1]. In this paper it is proved that the free abelian topological group on a compact Hausdorff space X generates the same variety if and only if X is not scattered.

Open subgroups of free abelian topological groups

1993 ◽  
Vol 114 (3) ◽  
pp. 439-442 ◽  
Author(s):  
Sidney A. Morris ◽  
Vladimir G. Pestov
Keyword(s):  
Topological Group ◽  
Sufficient Conditions ◽  
Open Subgroup ◽  
Regular Space ◽  
Topological Groups ◽  
Covering Dimension ◽  
Abelian Topological Group ◽  
Completely Regular ◽  
Free Topological Group ◽  
Free Abelian Topological Group

We prove that any open subgroup of the free abelian topological group on a completely regular space is a free abelian topological group. Moreover, the free topological bases of both groups have the same covering dimension. The prehistory of this result is as follows. The celebrated Nielsen–Schreier theorem states that every subgroup of a free group is free, and it is equally well known that every subgroup of a free abelian group is free abelian. The analogous result is not true for free (abelian) topological groups [1,5]. However, there exist certain sufficient conditions for a subgroup of a free topological group to be topologically free [2]; in particular, an open subgroup of a free topological group on a kω-space is topologically free. The corresponding question for free abelian topological groups asked 8 years ago by Morris [11] proved to be more difficult and remained open even within the realm of kω-spaces. In the present paper a comprehensive answer to this question is obtained.

EMBEDDING OF THE FREE ABELIAN TOPOLOGICAL GROUP INTO

Mathematika ◽  
2019 ◽  
Vol 65 (3) ◽  
pp. 708-718 ◽  
Author(s):  
Mikołaj Krupski ◽  
Arkady Leiderman ◽  
Sidney Morris
Keyword(s):  
Topological Group ◽  
Abelian Topological Group ◽  
Free Abelian Topological Group
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