A note on group invariant continua

The topological properties of "weak mixing" and "strong mixing" for topological transformation groups have been introduced and investigated and many analogous results have been derived. Let us adopt the following definitions and notations. We will restrict our attention to topological transformation groups (X, T) with compact metrizable phase space X. We call a topological transformation group ergodic if the only closed proper invariant subset is nowhere dense or, equivalently, there exists a point in X the orbit-closure of which is the whole space X. Such a point is called a transitive point. We call a topological transformation group weakly mixing if (X, T)x (X, T) is ergodic and strongly mixing if for any two open sets U and V, there is a compact subset K of T such that t ~ T ~ K implies Ut n V # ;J. Let ~'(X) be the set of all bounded complex-valued functions on X which are continuous on a co-meager set. (We identify functions which coincide on a co-meager set.) Let us define a cascade to be a topological transformation group with the discrete additive group of integers as phase group. In this situation, it is also denoted by (X, ~b), with ~b a homeomorphism of X. If Y is an invariant subset of X, then (Y, ~b) is also a cascade and we say that (Y, ~b) is a subcascade of (X, ~b). Let us consider the following properties of a topological transformation group (X, T):

1. Let (X, T, 7r) be a topological transformation group, where X is a nontrivial Hausdorff space and T is a topological group which leaves an end point e of X fixed. In [3], Wallace proved that if T is cyclic and X is a locally connected continuum, T has a fixed point other than e. Then in [4], Wallace asked the following question: If X is a peano continuum and T is compact or abelian, then does T have a fixed point other than e? In [5] Wang showed that if T is compact, and X is arcwise connected, then T has a fixed point other than e. Then Chu [1 ] showed that T has infinitely many fixed points. Chu began the study of the abelian case in [2]. In this paper we show by example that X may be a peano continuum and T may be a countably generated abelian group which has e for its only fixed point, ?2. Thus in general the answer to Wallace's question in the abelian case is no. However, if T is a generative group, and X is compact and arcwise connected, then T has a fixed point other than e, ?3. We also show by example that T may be a finitely generated nonabelian group which has e for its only fixed point, ?4.

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: Let G be a topological transformation group on a compact Hausdorff space Y and F(G;Y) its fixed point set. The analysis is devoted to the study of the cohomology structure of F(G;Y) in the following three cases: (1) G is the group Z2 of integers modulo 2 an Y has the mod 2 cohomology ring of the real projective n-space. (2) G is the group Zp of integers modulo p, where p is an odd prime number, and Y has the mod P cohomology structure of the lense ( <N+1)- space mod p. (3) G is the circle group S' and Y has the integral cohomology ring of the complex projective n-space. For simplicity, we shall call Y a cohomology real projective n-space or a cohomology lense (<N+1)-space mod p or a cohomology complex projective n-space if its cohomology structure is that described in (1 OR (2) or (3).

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.

1. In view of the recent work on topological groups it is natural to consider the situation which arises when such groups act as transformation groups on various types of spaces. Such a study is begun here from the point of view of almost periodic transformation groups, the definition of which is suggested by von Neumann's paper on almost periodic functions in a group.t Compact topological transformation groups are a special case of almost periodic transformation groups, at least for a rather wide class of spaces. The paper concerns itself chiefly with the nature of the minimal closed invariant sets of such groups. There are some results for general spaces but the main results are for Euclidean spaces and more particularly for threedimensional Euclidean spaces. One of the most interesting theorems states that if a compact one-dimensional group acts on three-space in such a way that its orbits are uniformly bounded in diameter, then every point of the space is fixed under the group, so that if such a group is to act in a nontrivial manner the diameters of its orbits must be unbounded. Under some restrictions a similar theorem is proved for one-parameter almost periodic groups. Furthermore it is shown that for this latter class of groups, many of the orbits must actually be simple closed curves if they have one-dimensional closures. 2. The group considered here will be denoted by G. It will be subjected to various conditions as the occasion demands but it will always be Abelian. In case it is the group of real numbers, it will be spoken of as a one-parameter group; in case it is the real numbers reduced modulo one, it will be spoken of as the circle group. The space on which the group acts will be denoted by R. It will be specialized in various ways, but in any case it will always be a locally compact metric space. If x and y are two points of R, the distance between them will be denoted by d(x, y).

Introduction. This article is based upon a principle which is so standard that it is almost a cliche: The first step to understanding a nonlinear phenomenon is to define and study a suitable linear approximation. To be more specific, we shall describe some applications of this to the symmetry questions of topological transformation groups. Given a topological space X, let Homeo(^) denote the set of self-homeomorphisms of X. This is a group under composition of mappings. If G is an arbitrary group, then a group action of G on I is a homomorphism Homeo( X). Frequently we wish to impose some weak assumptions on <p. For example, if G is a topological group, then we might want cp to have suitable continuity properties. The usual assumption is that the map

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

This book provides a general framework for doing geometric group theory for many non-locally-compact topological transformation groups that arise in mathematical practice, including homeomorphism and diffeomorphism groups of manifolds, isometry groups of separable metric spaces and automorphism groups of countable structures. Using Roe's framework of coarse structures and spaces, the author defines a natural coarse geometric structure on all topological groups. This structure is accessible to investigation, especially in the case of Polish groups, and often has an explicit description, generalising well-known structures in familiar cases including finitely generated discrete groups, compactly generated locally compact groups and Banach spaces. In most cases, the coarse geometric structure is metrisable and may even be refined to a canonical quasimetric structure on the group. The book contains many worked examples and sufficient introductory material to be accessible to beginning graduate students. An appendix outlines several open problems in this young and rich theory.

We establish a general framework for representability of a metric group on a (well-behaved) class of Banach spaces. More precisely, let G \mathcal {G} be a topological group, and A \mathcal {A} a unital symmetric C ∗ C^* -subalgebra of U C ( G ) \mathrm {UC}(\mathcal {G}) , the algebra of bounded uniformly continuous functions on G \mathcal {G} . Generalizing the notion of a stable metric, we study A \mathcal {A} -metrics δ \delta , i.e., the function δ ( e , ⋅ ) \delta (e, \cdot ) belongs to A \mathcal {A} ; the case A = W A P ( G ) \mathcal {A}=W\hskip -0.7mm A\hskip -0.2mm P(\mathcal {G}) , the algebra of weakly almost periodic functions on G \mathcal {G} , recovers stability. If the topology of G G is induced by a left invariant metric d d , we prove that A \mathcal {A} determines the topology of G \mathcal {G} if and only if d d is uniformly equivalent to a left invariant A \mathcal {A} -metric. As an application, we show that the additive group of C [ 0 , 1 ] C[0,1] is not reflexively representable; this is a new proof of Megrelishvili [ Topological transformation groups: selected topics , Elsevier, 2007, Question 6.7] (the problem was already solved by Ferri and Galindo [Studia Math. 193 (2009), pp. 99–108] with different methods and later the results were generalized by Yaacov, Berenstein, and Ferri [Math. Z. 267 (2011), pp.129–138]). Let now G \mathcal {G} be a metric group, and assume A ⊆ L U C ( G ) \mathcal {A}\subseteq \mathrm {LUC}(\mathcal {G}) , the algebra of bounded left uniformly continuous functions on G \mathcal {G} , is a unital C ∗ C^* -algebra which is the uniform closure of coefficients of representations of G \mathcal {G} on members of F \mathscr {F} , where F \mathscr {F} is a class of Banach spaces closed under ℓ 2 \ell _2 -direct sums. We prove that A \mathcal {A} determines the topology of G \mathcal {G} if and only if G \mathcal {G} embeds into the isometry group of a member of F \mathscr {F} , equipped with the weak operator topology. As applications, we obtain characterizations of unitary and reflexive representability.

1. Introduction.Let S be a locally compact semigroup with a dense maximal subgroup G.We ask the following question: When is G a topological group under the induced topology?It was proved recently by P. S. Mostert [8] that in general G is not a topological group.However, there are several good reasons motivating the study of such a problem.First, for an arbitrary locally compact semigroup, the closure of the maximal subgroup containing an idempotent forms a locally compact semigroup.If we know this maximal subgroup is a topological group, then it is a locally compact topological group.(This fact will be proved in the following paragraph.)By our rather good knowledge of locally compact topological groups, we can obtain information about the structure of this locally compact semigroup.Second, if we look at this problem from the point of view of transformation groups, we have a group G acting on a locally compact topological space.Naturally, we would like this group to be topologized so that it is a topological group.It is known that the weakest topology for G to be a topological group is the g-topology [1].It is somewhat harder to deal with the g-topology than with the fc-topology, but under some conditions the ^-topology will coincide with the fe-topology.Among the known conditions, the local compactness of the group seems most interesting.If one tries to generalize such a condition, it is rather natural to ask the following : If G is a group of homeomorphisms of X onto X, and the closure of G is locally compact in the fc-topology, does the induced topology of G coincide with the g-topology?Third, in the past few years there have been several papers about such questions : What is the relationship between the algebraic structure and the topological structure?When can one strengthen the topological structure so that even under weak conditions the algebraic operations will be continuous in a certain sense?For instance, in topological groups, if the group space is locally compact, and if the multiplication is separately continuous with respect to the topology of the space, then the group is a topological group.We are able to prove the same theorem when the space is second countable and second category without the assumption of local compactness [9].Now, our present problem arises quite

Retraction properties of the orbit space of a compact topological transformation group

n this dissertation, we present new set functions called strongly limit and strongly prolongation limit sets. We show the new sets, especially strongly prolongation limit sets, characterize proper action under an arbitrary setting. That is, we characterize proper action for wider class of proper ܩ-spaces. Also, we show the new version of the sets could be derived from strongly exceptional sets which have been used as a good technique for the characterization of a proper maps. Moreover, we review properties of well-known limit sets and prolongations and properties for the new version of limit sets under an arbitrary setting on a ܩ-space ܺ. Next, we give characterizations for a proper action and a locally proper action using the new set functions and some relevant areas where one could use these sets to describe some topological properties for a space ܺ or algebraic properties for a group ܩ acting on ܺ. Following the technique of limit sets, we present a new version of action as weaker forms of the proper action called ݓ -actions using convergence techniques. This approach yields new directions for the proof of some existing theorems which are then given as well as come counter-examples. We give some applications for our weaker forms and how convergence techniques could be fruitful to explain the relations among certain notions in the theory of ܩ-spaces. Finally, according to our weaker forms, we characterize the notion of enough slice that has presented by H. Biller using ݓ -Cartan action.

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.

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.