Subgroups and subrings of profinite rings

1994 ◽  
Vol 116 (2) ◽  
pp. 209-222 ◽  
Author(s):  
A. G. Abercrombie
Keyword(s):  
Topological Group ◽  
Hausdorff Dimension ◽  
Haar Measure ◽  
Arbitrary Dimension ◽  
Invariant Probability Measure ◽  
Fractional Dimension ◽  
Invariant Metric ◽  
Analogous Statement ◽  
Natural Choice ◽  
Carry Over

AbstractA profinite topological group is compact and therefore possesses a unique invariant probability measure (Haar measure). We shall see that it is possible to define a fractional dimension on such a group in a canonical way, making use of Haar measure and a natural choice of invariant metric. This fractional dimension is analogous to Hausdorff dimension in ℝ.It is therefore natural to ask to what extent known results concerning Hausdorff dimension in ℝ carry over to the profinite setting. In this paper, following a line of thought initiated by B. Volkmann in [12], we consider rings of a-adic integers and investigate the possible dimensions of their subgroups and subrings. We will find that for each prime p the ring of p-adic integers possesses subgroups of arbitrary dimension. This should cause little surprise since a similar result is known to hold in ℝ. However, we will also find that there exists a ring of a-adic integers possessing Borel subrings of arbitrary dimension. This is in contrast with the situation in ℝ, where the analogous statement is known to be false.

scholarly journals Haar measure and compact right topological groups

1992 ◽  
Vol 45 (3) ◽  
pp. 399-413 ◽  
Author(s):  
Paul Milnes
Keyword(s):  
Topological Group ◽  
Probability Measure ◽  
Haar Measure ◽  
Structure Theorem ◽  
Invariant Probability Measure ◽  
Sufficient Conditions ◽  
Positive Answer ◽  
Topological Groups ◽  
Low Dimensional ◽  
Compact Right Topological Group

The consideration of compact right topological groups goes back at least to a paper of Ellis in 1958, where it is shown that a flow is distal if and only if the enveloping semigroup of the flow is such a group (now called the Ellis group of the distal flow). Later Ellis, and also Namioka, proved that a compact right topological group admits a left invariant probability measure. As well, Namioka proved that there is a strong structure theorem for compact right topological groups. More recently, John Pym and the author strengthened this structure theorem enough to be able to establish the existence of Haar measure on a compact right topological group, a probability measure that is invariant under all continuous left and right translations, and is unique as such. Examples of compact right topological groups have been considered earlier. In the present paper, we give concrete representations of several Ellis groups coming from low dimensional nilpotent Lie groups. We study these compact right topological groups, and two others, in some detail, paying attention in particular to the structure theorem and Haar measure, and to the question: is Haar measure uniquely determined by left invariance alone? (It is uniquely determined by right invariance alone.) To assist in answering this question, we develop some sufficient conditions for a positive answer. We suspect that one of the examples, a compact right topological group coming from the Euclidean group of the plane, does not satisfy these conditions; we don't know if the question has a positive answer for this group.

scholarly journals On Theorems of Kawada and Wendel

1958 ◽  
Vol 11 (2) ◽  
pp. 71-77 ◽  
Author(s):  
J. H. Williamson
Keyword(s):  
Topological Group ◽  
Haar Measure ◽  
Borel Measure ◽  
Convolution Product ◽  
Locally Compact ◽  
Compact Topological Group ◽  
Complex Functions ◽  
Image Position ◽  
Locally Compact Topological Group ◽  
Invariant Haar Measure

Let G be a locally compact topological group, with left-invariant Haar measure. If L1(G) is the usual class of complex functions which are integrable with respect to this measure, and μ is any bounded Borel measure on G, then the convolution-product μ⋆f, defined for any f in Li byis again in L1, and

scholarly journals Dimension, entropy and Lyapunov exponents

1982 ◽  
Vol 2 (1) ◽  
pp. 109-124 ◽  
Author(s):  
Lai-Sang Young
Keyword(s):  
Hausdorff Dimension ◽  
Probability Measure ◽  
Lyapunov Exponents ◽  
Borel Probability Measure ◽  
Fractional Dimension ◽  
Present Context ◽  
Rényi Dimension ◽  
Borel Probability

AbstractWe consider diffeomorphisms of surfaces leaving invariant an ergodic Borel probability measure μ. Define HD (μ) to be the infimum of Hausdorff dimension of sets having full μ-measure. We prove a formula relating HD (μ) to the entropy and Lyapunov exponents of the map. Other classical notions of fractional dimension such as capacity and Rényi dimension are discussed. They are shown to be equal to Hausdorff dimension in the present context.

scholarly journals SINGULARITIES OF THE BIEXTENSION METRIC FOR FAMILIES OF ABELIAN VARIETIES

2018 ◽  
Vol 6 ◽  
Author(s):  
JOSÉ IGNACIO BURGOS GIL ◽  
DAVID HOLMES ◽  
ROBIN DE JONG
Keyword(s):  
Abelian Varieties ◽  
Arbitrary Dimension ◽  
Invariant Metric ◽  
Hodge Structures

In this paper we study the singularities of the invariant metric of the Poincaré bundle over a family of abelian varieties and their duals over a base of arbitrary dimension. As an application of this study we prove the effectiveness of the height jump divisors for families of pointed abelian varieties. The effectiveness of the height jump divisor was conjectured by Hain in the more general case of variations of polarized Hodge structures of weight $-1$.

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.

Convolution in perfect Lie groups

2016 ◽  
Vol 161 (1) ◽  
pp. 31-45
Author(s):  
YVES BENOIST ◽  
NICOLAS DE SAXCÉ
Keyword(s):  
Hausdorff Dimension ◽  
Lie Groups ◽  
Haar Measure ◽  
Lie Group ◽  
Borel Measure ◽  
Absolutely Continuous ◽  
Borel Set ◽  
Open Set ◽  
Convolution Power ◽  
Smooth Density

AbstractLetGbe a connected perfect real Lie group. We show that there exists α < dimGandp∈$\mathbb{N}$* such that if μ is a compactly supported α-Frostman Borel measure onG, then thepth convolution power μ*pis absolutely continuous with respect to the Haar measure onG, with arbitrarily smooth density. As an application, we obtain that ifA⊂Gis a Borel set with Hausdorff dimension at least α, then thep-fold product setApcontains a non-empty open set.

Transformations and Colorings of Groups

2007 ◽  
Vol 50 (4) ◽  
pp. 632-636 ◽  
Author(s):  
Yevhen Zelenyuk ◽  
Yuliya Zelenyuk
Keyword(s):  
Topological Group ◽  
Haar Measure ◽  
Continuous Transformation ◽  
Compact Topological Group

AbstractLet G be a compact topological group and let f : G → G be a continuous transformation of G. Define f* : G → G by f*(x) = f (x–1)x and let μ = μG be Haar measure on G. Assume that H = Im f* is a subgroup of G and for every measurable C ⊆ H, μG(( f*)–1(C)) = μH(C). Then for every measurable C ⊆ G, there exist S ⊆ C and g ∈ G such that f (Sg–1) ⊆ Cg–1 and μ(S) ≥ (μ(C))2.

On critical pairs of product sets in a certain matrix group

1970 ◽  
Vol 67 (3) ◽  
pp. 569-581
Author(s):  
M. McCrudden
Keyword(s):  
Topological Group ◽  
Haar Measure ◽  
Finite Measure ◽  
Matrix Group ◽  
Real Numbers ◽  
Locally Compact ◽  
Image Position ◽  
Set Function ◽  
Product Sets ◽  
Critical Pairs

1. Introduction. If G is a locally compact Hausdorff topological group, and μ is (left) Haar measure on G, then we denote by ℬ(G) the class of all Borel subsets of G having finite measure, and by VG the set {μ(E): E ∊ ℬ(G)} of real numbers. The product set function of G, ΦG: VG × VG → VG, is defined (see (4) and (5)) byand, for each u, v ∈ VG, we call a pair (E, F) of Borel subsets of G a critical (u, v)-pair, if μ(E) = u, μ(F) = v, and μ*(EF) = ΦG(u, v). We denote the class of all critical (u, v)-pairs by and we write ℰG for .

Lp(G)-isotone measures

1975 ◽  
Vol 78 (3) ◽  
pp. 471-481 ◽  
Author(s):  
Beryl J. Peers
Keyword(s):  
Topological Group ◽  
Haar Measure ◽  
Equivalence Classes ◽  
Locally Compact ◽  
Compact Topological Group ◽  
Borel Measures ◽  
Integrable Functions ◽  
Locally Compact Topological Group

Let G be a locally compact topological group with left Haar measure, m; let M(G) denote the bounded regular Borel measures on G and let Lp(G) denote the equivalence classes of pth power integrable functions on G with respect to the left Haar measure.

scholarly journals Conformal Invariance of CLEΚ on the Riemann Sphere for Κ ∈(4,8)

2020 ◽  
Author(s):  
Ewain Gwynne ◽  
Jason Miller ◽  
Wei Qian
Keyword(s):  
Invariant Measure ◽  
Probability Measure ◽  
Entire Range ◽  
Riemann Sphere ◽  
Invariant Probability Measure ◽  
Intermediate Step ◽  
Analogous Statement ◽  
Conformally Invariant ◽  
Simply Connected Domain ◽  
Simply Connected

Abstract The conformal loop ensemble (${\textrm{CLE}}$) is the canonical conformally invariant probability measure on non-crossing loops in a simply connected domain in $\mathbbm{C}$ and is indexed by a parameter $\kappa \in (8/3,8)$. We consider ${\textrm{CLE}}_\kappa $ on the whole-plane in the regime in which the loops are self-intersecting ($\kappa \in (4,8)$) and show that it is invariant under the inversion map $z \mapsto 1/z$. This shows that whole-plane ${\textrm{CLE}}_\kappa $ for $\kappa \in (4,8)$ defines a conformally invariant measure on loops on the Riemann sphere. The analogous statement in the regime in which the loops are simple ($\kappa \in (8/3,4]$) was proven by Kemppainen and Werner and together with the present work covers the entire range $\kappa \in (8/3,8)$ for which ${\textrm{CLE}}_\kappa $ is defined. As an intermediate step in the proof, we show that ${\textrm{CLE}}_\kappa $ for $\kappa \in (4,8)$ on an annulus, with any specified number of inner-boundary-surrounding loops, is well defined and conformally invariant.

Sign in / Sign up

Export Citation Format

Share Document