Topological Transformation Group Research Articles

Isotropy group on some topological transformation group structures

This paper explores the topological properties of irresolute topological groups, their quotient maps, and the role of topology in normal subgroups. It provides a detailed analysis\linebreak using examples and counterexamples. The study focuses on the essential features of irresolute topological groups and their quotient groups, for understanding the topological aspects of isotropy groups. For a trans\-for\-ma\-tion group $(\mathsf{H}, \mathsf{Y}, \psi)$ and a point $y \in \mathsf{Y},$ the set \centerline{$\mathsf{H}_{y} = \{h \in \mathsf{H} \colon hy = y\}$} \noi consisting of elements of $\mathsf{H}$ that fix $y$, is called the isotropy group at $y$. The paper highlights the distinct topological characteristics of isotropy groups in transformation group structure. It demonstrates that if $(\mathsf{H}, \mathsf{Y}, \psi)$ is an Irr$^{*}$-topological transformation group, then $( \mathsf{H}/ \mathop{Ker} \psi, \mathsf{Y}, \overline{\psi})$ forms an effective Irr$^{*}$-topological transformation group. By investigating both irresolute topological groups and isotropy groups, the study provides a clear understanding of their topological features. This research improves our understanding of these groups by offering clear examples and counterexamples, leading to a thorough conclusion about their different topological features.

On a New Type of Topological Transformation Group

On a New Type of Topological Transformation Group

Syndetic proximal set in topological transformation group

In this paper will study the syndetic set in topological transformation group. It represents one of the most important sets in topological dynamics that Gottschak referred .We Introduce dynamical relationship between syndetic set and translation (left-right) ,In topological dynamics group present semi replet set and extensive set .We referred to the relationship of semi replet set and extensive set on the one hand, and their relationship to syndetic set on the other hand . We introduced a new concept syndetic proximal point that depended on syndetic set we indicated a relationship syndetic proximal point with proximal point and extensive proximal point , conclude a number of relationships and concepts

On The Minimal Sets and Stability Conditions For Compact Sets In I(X)-spaces

The set of all isometries on a metric space X with the usual composition of functions form a group and it is called the group of isometries and is denoted by I(X). In this paper we study the generalization of the concepts of minimal sets, stability and attraction, from dynamic system into the topological transformation group (I(X),X).We find that the collection of all minimal sets of I(X)-space is the collection of all the closures of orbits of X and we found some useful results about stability and attraction and we fixed the relationship among it's kinds.

A topological transformation group without non-trivial equivariant compactifications

There is a countable metrizable group acting continuously on the space of rationals in such a way that the only equivariant compactification of the space is a singleton. This is obtained by a recursive application of a construction due to Megrelishvili, which is a metric fan equipped with a certain group of homeomorphisms. The question of existence of a topological transformation group with the property in the title was asked by Yu.M. Smirnov in the 1980s.

On Mixing Properties in Topological Dynamics

   Mixing property was classified in the first book of topological dynamics written by Gottschalk & Hedlund ,[4], ( 1955 ) as one of the regionally transitive properties . It is concerned with the existence of a compact subset  of the phase group   of a topological transformation group  such that for every  non empty open subsets of ,there exists a compact subset  of    such that, for every        In this work we introduced new different types of mixing ,and  studied there relations with other transitive properties. Moreover  we studied some relations between them and recursion dynamical properties such as periodicity , almost periodicity and recurrence. we gave conditions to transfer some types of mixing under suitable homomorphism.Â

When the cone in Bochner's theorem has an opening less than π

We prove that if the Fourier transform of a measure μ∈M(ℝn) vanishes in the interior of a cone of opening less than π, then the distance set {|x−τ|: x∈G∩supp μ} has a positive one-dimensional Lebesgue measure whenever τ∈ℝn, G⊂ℝn is open, and μ(G)≠0. When the cone has an opening greater than π, it is well known that μ is absolutely continuous with respect to Lebesgue measure on ℝn. For measures on the n-torus, this dates back to a 1944 paper of Bochner. For measures on ℝn, this follows from a Euclidean space version of Forelli's theorem about analytic and quasi-invariant measures. Forelli's theorem was proved in a 1967 paper in the abstract setting of a locally compact Hausdorff space on which ℝ acts as a topological transformation group. We give a Euclidean space proof of this Euclidean space version of Forelli's theorem.

Minimal sets and stability For Compact sets of I(X)-spaces

The set of all isometries on a metric space X with the usual composition of functions form a group and it is called the group of isometries and is denoted by I(X). In this paper we study the generalization of the concepts of minimal sets, stability and attraction, from dynamic system into the topological transformation group (I(X),X).We find that the collection of all minimal sets of I(X)-space is the collection of all the closures of orbits of X and we found some useful results about stability and attraction and we fixed the relationship among it's kinds.

Cantor Limit Set of a Topological Transformation Group on S1

The limit set of a topological transformation group on S1 generated by two generators is proved to be totally disconnected (or thin) and perfect if the conditions (i–v) are satisfied. A concrete application to a Doubly Periodic Riccati equation is given.

Almost periodicity and equicontinuity of the topological transformation group

In this paper, a characterization of almost periodicity of topological transformation groups on uniform spaces is given. By searching the appropriate base for uniform structure, it is shown that the topological transformation group is topologically equivalent to an isometric one if it is uniformly equicontinuous.

Semi-exactitude du bifoncteur de Kasparov pour les actions moyennables

Consider an equivariant extension of graded separable G-algebras which admits a completely linear positive, grading preserving cross section (not necessary equivariant) of norm 1. We denote ( X , G ) an amenable topological transformation group in the sense of Anantharaman-Delaroche. We establish an isomorphism concerning the Kasparov equivariant bifunctor R KK G ( X ; − , − ) . This isomorphism in K-theory, allows one to extend the half-exactness from the case of the proper algebras (which is analogue to the one obtained by Skandalis in the non-equivariant case) to the case of amenable actions. In particular, we will place ourselves in a significant case, that of hyperbolic displacements of the Poincaré–Lobatschevsky geometry on the unit disc. To cite this article: D. El Morsli, C. R. Acad. Sci. Paris, Ser. I 341 (2005).

Local connectedness in transformation groups

This note shows that under very general conditions, in a topological transformation group, the natural map from the group onto an orbit is almost open. The implications for local connectedness are investigated. In particular, if the image of a path component of the group is sufficiently “robust", the orbit will be locally connected.

HILBERT'S FIFTH PROBLEM: REVIEW

Among the twenty-three mathematical problems that Hilbert posed at the Second International Congress of Mathematicians in Paris in 1900 (see [10]), the fifth problem is concerned with Lie’s theory of transformation groups, and in its second part deals with what Hilbert calls “infinite groups,” but which are not groups in the modern use of the term. The questions in this second part of the fifth problem concern functional equations and difference equations and have, for example, connections with the work of N.H. Abel. These questions lie completely outside the theory of transformation groups, and we do not discuss them here any further. We refer the reader to [1] for the “state of the art” of 1989 in this area of research. We recall that a topological transformation group consists of a topological group G, a topological space X, and a continuous action of G on X, that is, a continuous map

Transforming random elements and shifting random fields

Consider a locally compact second countable topological transformation group acting measurably on an arbitrary space. We show that the distributions of two random elements X and $X'$ in this space agree on invariant sets if and only if there is a random transformation $\Gamma$ such that $\Gamma X$ has the same distribution as $X'$. Applying this to random fields in d dimensions under site shifts, we show further that these equivalent claims are also equivalent to site-average total variation convergence. This convergence result extends to amenable groups.

A sufficient condition for the existence of a G-compactification

In this paper we show that if a topological transformation group is equicontinuous with respect to some uniformity for the phase space, then the topological transformationgroup can equivariantly be embedded in a topological transformationgroup with compact phase space.

Group Extensions and Cohomology for Locally Compact Groups. III

We shall define and develop the properties of cohomology groups ${H^n}(G,A)$ which can be associated to a pair (G, A) where G is a separable locally compact group operating as a topological transformation group of automorphisms on the polonais abelian group A. This work extends the results in [29] and [30], and these groups are to be viewed as analogues of the Eilenberg-Mac Lane groups for discrete G and A. Our cohomology groups in dimension one are classes of continuous crossed homomorphisms, and in dimension two classify topological group extensions of G by A. We characterize our cohomology groups in all dimensions axiomatically, and show that two different cochain complexes can be used to construct them. We define induced modules and prove a version of Shapiro’s lemma which includes as a special case the Mackey imprimitivity theorem. We show that the abelian groups ${H^n}(G,A)$ are themselves topological groups in a natural way and we investigate this additional structure.

Group extensions and cohomology for locally compact groups. III

We shall define and develop the properties of cohomology groups H n ( G , A ) {H^n}(G,A) which can be associated to a pair (G, A) where G is a separable locally compact group operating as a topological transformation group of automorphisms on the polonais abelian group A. This work extends the results in [29] and [30], and these groups are to be viewed as analogues of the Eilenberg-Mac Lane groups for discrete G and A. Our cohomology groups in dimension one are classes of continuous crossed homomorphisms, and in dimension two classify topological group extensions of G by A. We characterize our cohomology groups in all dimensions axiomatically, and show that two different cochain complexes can be used to construct them. We define induced modules and prove a version of Shapiro’s lemma which includes as a special case the Mackey imprimitivity theorem. We show that the abelian groups H n ( G , A ) {H^n}(G,A) are themselves topological groups in a natural way and we investigate this additional structure.

Universal topological transformation groups

Let G denote an infinite locally compact topological group. In this paper a topological transformation group is constructed in which all topological transformation groups( G, X, π) can be embedded for which X is a completely regular space of weight w( X) ⩽ the Lindelöf degree L( G) of G. As an application, results of de Groot and of de Groot and McDowell are generalized concerning the extension of the action of a group to a completion, resp. a compactification, of the phase space.

A universal topological transformation group inL 2 (G × G)

LetG denote an infinite,σ-compact, locally compact topological group. In this paper a construction is given for a topological transformation groupHG with the Hilbert spaceL 2 (G × G) as a phase space such that any topological transformation group (G, X, π) can be embedded inH G , providedX is a separable metrizable space andπ is a bounded action. The class of such topological transformation groups contains all actions ofG on separable, metrizable, locally compact spaces.

On low dimensional minimal sets

Let (X, G, f) be a topological transformation group. Suppose that the phase space X is compact, separable metric, and locally contractible and the group G is the additive group of all real numbers R with the usual topology. If X is a minimal set of diπii (X) ^ 2 then X is a manifold, imposing a further condition on the action when dimz (X) = 2. Hence X is a singleton, a circle or a torus according to its dimension.