Topological Transformation Groups Research Articles

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 the geometric weight system of topological actions on cohomology quaternionic projective spaces

In this paper, we take those cohomology manifolds X of rational cohomology type of quaternionic projective spaces as the testing spaces for the study of cohomology theory of topological transformation groups. We follow the general viewpoint set out in [-3, 4] of seeking for the socalled topological Schur lemma in the form of splitting theorem in equivariant cohomology for torus group actions on cohomology quaternionic projective spaces. The main result of this paper is precisely such a structural theorem, Theorem(2.1), which proves that topological actions of torus groups (of rank >__ 2) have exactly the same cohomological behavior as that of suitable linear models. Namely, for a cohomology manifold X of QP-type with a given effective action of a torus group G (rank _>2, or 3 if all fixed points are isolated), there exists a unique lifting ( of the generator (o of H * ( X ) into H G (X) such that

Construction and properties of space-timeb-boundaries

A criterion is given for two curves of finite generalized affine length to define the same point of the b-boundary of a space-time. It is shown that the b-boundary can be constructed on every closed subbundle of the bundle of linear frames to which the Levi-Civita connection is reducible. It follows that, for any product space-time, the space-time together with its b-boundary is homeomorphic to a product of pseudo-Riemannian spaces with b-boundary. Furthermore, it is shown that maps of one space-time in to another which are isomorphisms of the connections can be C0-extended to the space-time with b-boundary. In particular, it follows that the group of affine transformations and the group of isometries of a space-time act as topological transformation groups on the spacetime with b-boundary.

On weak mixing of cascades

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):

Some fundamental theorems in cohomology theory of topological transformation groups

Some fundamental theorems in cohomology theory of topological transformation groups

Almost periodicity and B-equicontinuity in topological dynamics

In the previous paper(8), we considered a property of families of functions we termed. ‘B-equicontinuity’. It was shown that B-equicontinuity is stronger than the usual equicontinuity, and is weaker than the equicontinuity defined by Bartle (3). In this paper we consider the concept of B-equicontinuity on topological transformation groups. The net characterization of equicontinuity obtained in (8) is used in discussion. It is proved in (1) that if (X, T, π) is almost periodic, the transition group {πt|t ∈ T} is equicontinuous. One might wonder whether this conclusion can be strengthened to say that {πt|t ∈ T} is B-equicontinuous; we show here by an example that this is not true and a partial solution to this problem is given. Some relations between almost periodicity and B -equicontinuity are also discussed.

On topological transformation groups

The purpose of this note is to prove that under certain conditions, if a compact Lie group G acts on a space M, then there will exist uncountably many topologically distinct actions of G on M. Montgomery and Yang have recently shown in [3] that SI can act on S7 in countably many topologically distinct ways. Their methods are different from the ones employed in this paper. In both cases, the actions are distinguished by the orbit spaces. However, Montgomery and Yang distinguish the orbit spaces, which are in fact differentiable manifolds, by global properties. In this note, the orbit spaces are distinguished by local properties, and hence they are not manifolds. The notation and terminology are standard. If G acts on M, and x is an element of M, then G. will denote the isotropy subgroup, G/G. the quotient subgroup, G(x) the orbit of x, M/G the orbit space, and PG the projection map from M to M/G. If A is an arc in the interior of a k-cell D, then D/A will denote the space obtained by identifying A to a point.

On ergodicity and mixing in topological transformation groups

On ergodicity and mixing in topological transformation groups

Topological Transformation Groups with a Fixed End Point

Topological Transformation Groups with a Fixed End Point

Topological transformation groups with a fixed end point

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.

A note on topological transformation groups with a fixed end point

A note on topological transformation groups with a fixed end point

Homomorphisms of topological transformation groups into function spaces

Homomorphisms of topological transformation groups into function spaces

Equicontinuity and -recursive transformation groups

In this paper we obtain results on equicontinuity and apply them to certain recursive properties of topological transformation groups (X, T, π) with uniform phase space X. For example, in the special case that each transition πt is uniformly continuous we consider the transformation group (X,Ψ,ρ), where Ψ is the closure of {πt|t∈T} in the space of all unimorphisms of X onto X with the topology of uniform convergence (space index topology) and p(x, φ) = φ(x) for (x, φ)∈ X × Ψ. (See (1), page 94, 11·18.) If π is a mapping on X × T we usually write ‘xt’ for ‘π(x, t)’ and ‘AT’ for ‘π(A × T)’ where A ⊂ X. In this case we obtain the following results:I. If (X, T, π) is almost periodic [regularly almost periodic] and πxis equicontinuous, then (X,Φ,ρ) is almost periodic [regularly almost periodic]II. Let A be a compact subset of X such that. If the left and right uniformities of T are equal and (X, T, π) is almost periodic [regularly almost periodic], then (X, T, π) is almost periodic [regularly almost periodic].

Transformation groups in the theory of topological loops

In a recent paper [5] the author has applied the techniques of topological transformation group theory to the study of certain topological loops. The purpose of this note is to show more explicitly the close connection between topological loops and topological transformation groups. It is shown that for every transformation group G which acts reasonably on a space X so that (a) there is a global cross-section a from the coset space GIG, into G for pEX, (b) 5(G/G,) is a strongly transitive collection of homeomorphisms of X, and (c) G, is compact, such a space X may be given a binary operation so that it becomes a topological loop with a left invariant uniformity. Conversely, it is shown that in certain cases a topological loop L allows a transformation group G to act reasonably on L with a cross-section from G/G1 for the identity 1 of L satisfying (a) and (b) above. In certain cases (c) will also be satisfied. The reader may consult K. H. Hofmann [3] for the appropriate terminology in topological loops, with the following exceptions. A topological loop L has a left invariant uniformity (see [5 ]) when there is a uniform structure 'IU on L compatible with its topology so that 'I has a base a of entourages satisfying (x, y) EB if and only if (ax, ay) EB for all BE 6 and a EL. An invariant uniformity is a left invariant uniformity which also satisfies a right invariant condition. If a and b are elements of a loop, the unique solutions x and y to the equations ax =b and ya = b will be designated in this note by a(-I)b and ba(-1), respectively. The parentheses embracing -1 are used as a reminder that a(1)6 is not in general the product of 'a- and b as in a group. (This is at least the third notation appearing in print for these solutions. At the present this seems to be the most satisfactory.) The reference for transformation group topics is Montgomery and Zippin [7]. If G is a transformation group acting on a space X, and if G, is the isotropy or stability group at pEX, then there is a canonical mapping 7r: G/Gp--X defined by r(gG.) =g(p). Whenever ir is a homeomorphism, G is said to act reasonably on X (see [8]). The canonical mapping Xis continuous, onto, and one-one when G acts effectively and transitively [7, p. 43], but not in general open. A cross-section from G/G, into G is a continuous mapping a of the coset

An extension of Banach’s contraction principle

1. A. Borel, Seminar on transformation groups, Annals of Mathematics Studies No. 46, Princeton, 1960. 2. G. E. Bredon, Frank Raymond, R. F. Williams, p-adic groups of transformations, to appear in Trans. Amer. Math. Soc. 3. D. Montgomery and L. Zippin, Topological transformation groups, New YorkLondon, Interscience, 1955. 4. Frank Raymond, Union and separation theorems for generalized manifolds with boundary, Michigan Math. J. vol. 7 (1960) pp. 7-21. 5. G. T. Whyburn, Analytic topology, Amer. Math. Soc. Colloquim Publications, vol. 28, 1942. 6. R. L. Wilder, Monotone mappings of manifolds. II, Michigan Math. J. vol. 5 (1958) pp. 19-23. 7. C. T. Yang, p-adic transformation groups, Michigan Math. J. vol. 7 (1960) pp. 201-218.

Locally compact transformation groups

In ?1 of this paper it is shown that a variety of conditions implying nice behavior for topological transformation groups are, in the presence of separability, equivalent. In ?2 the continuity properties of the stability subgroups are studied. The conditions of ?1 exclude the line acting on the torus in such a way that each orbit is dense. They exclude the integers acting on the circle by rotation through multiples of an irrational angle and they exclude the group of those sequences of zeros and ones which have all but a finite number of their terms equal to zero when this group acts on the space of all sequences of zeros and ones by coordinatewise addition (mod 2). As we shall see in the proof of Theorem 1, the latter transformation group is a prototype for all excluded transformation groups. This is analogous to the following fact in the theory of Rings of Operators: Every factor of type II, contains a hyperfinite factor of type II,. The conditions were suggested by [3, Theorem 1] and the proof of their equivalence is somewhat analogous to the proof of [3, Theorem 1]. However, the proof does not depend upon [3] nor upon the theory of C*-algebras.

Topological Transformation Groups. By D. Montgomery and L. Zippin (Interscience tracts in pure and applied mathematics No. 1). pp. 281. $5.50. 1955. (Interscience Publishers Inc., New York & London)

Topological Transformation Groups. By D. Montgomery and L. Zippin (Interscience tracts in pure and applied mathematics No. 1). pp. 281. $5.50. 1955. (Interscience Publishers Inc., New York & London) - Volume 41 Issue 336

Book Review: Topological transformation groups

Book Review: Topological transformation groups

Asymptoticity in topological dynamics

This paper treats the notion of asymptoticity in topological transformation groups. By a topological transformation group we shall mean a topological group (the phase group) of homeomorphisms of a topological space (the phase space) onto itself, which contains the identity transformation as group identity and which has the property that any two homeomorphisms applied successively yield the same result as their group product applied once. The notion of asymptoticity has been discussed in the past only in the setting where the phase group was either the integers or the one-parameter group of reals (dynamical systems). In such systems two distinct points are said to be positively asymptotic provided their orbits under the set of positive integers (or reals), i.e., under a so-called positive ray, as homeomorphisms become and remain as close as one chooses. A similar notion is available for negative asymptoticity, i.e., asymptoticity under a negative ray; and if a pair of points is both positively and negatively asymptotic, they are said simply to be asymptotic. It is the purpose of this paper to extend the notion of asymptoticity to transformation groups with a more general structure. In ?1 we give preliminary definitions, which in the form here presented are due to Gottschalk and Hedlund [3](2). In ?2 we define an extended notion of asymptoticity relative to a replete semigroup, i.e., a semigroup of the phase group which contains a translate of each compact set in the phase group. A generalization of aand co-limit sets is available to us from [3, sec. 6.33] and is called the P-limit set of a point, where P refers to a replete semigroup. We restrict our attention largely to separable generative phase groups and prove a number of theorems about P-asymptoticity. It is shown that the P-limit set of a point has a minimality property in that it is the smallest set to which the point can be P-asymptotic, and further that the point actually is P-asymptotic to its P-limit set if and only if it does not belong to its P-limit set. The principal result of this section is a theorem about unstable transformation groups which states that under suitable hypotheses the existence of P-asymptotic points is guaranteed. This theorem contains as a special case a theorem due to Utz [6]. In ?3 we consider a symbolic transformation group, which is an extension

Topological Transformation Groups. I

Topological Transformation Groups. I