A study of the proximal relation in coset transformation groups

Introduction. Let (X,T) be a transformation group whose phase space X is a uniform space. Most of the results are stated for compact X. In this paper, four proximity relations, L, M, P, and Q, in X are defined and some of their properties studied. The relations P and Q were first defined and studied by Ellis and Gottschalk [5]. As a general reference for the notions occurring here, consult [8]. In Theorem 1, L is shown to be an invariant equivalence relation in X, and in Theorem 3, L is characterized as the union of all orbit closures under (X x X, T) which are contained in P. Theorem 3 also shows that if P is closed, then P = L and P is a closed invariant equivalence relation in X. Theorem 4 establishes the productivity of the relations Land M, and in Theorem 7, it is shown that (X,T) is coterminous, i.e., P = Q = L= M, iff P = Q. Theorems 9 through 13 describe L, M, P, and Q under various hypotheses such as (X,T) distal; (X,T) minimal and T abelian; (X, T) regionally mixing; and the like. Theorem 14 is an application of the general theory to obtain a characterization of (X,T) being uniformly equicontinuous, and as such represents a strengthening of a theorem by John D. Baum [2]. The author is indebted to Professor W. H. Gottschalk for his invaluable suggestions, mathematical stimulation, and sustained interest. STANDING NOTATION. Let (X,T) be a transformation group where X is always a uniform space. Let / be the class of all syndetic subsets of T, let XV be the class or all compact subsets of T, let OW be the uniformity of X, and for each x E X let

Higher order regionally proximal equivalence relations for general minimal group actions

Proximity relations in the fuzzy relational database model

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

Introduction.Let iX,T) be a transformation group whose phase space X is a uniform space.Most of the results are stated for compact X.In this paper, four proximity relations, L, M, P, and Q, in X are defined and some of their properties studied.The relations P and Q were first defined and studied by Ellis and Gottschalk [5].As a general reference for the notions occurring here, consult [8].In Theorem 1, Lis shown to be an invariant equivalence relation in X, and in Theorem 3, L is characterized as the union of all orbit closures under (X x X, T) which are contained in P. Theorem 3 also shows that if P is closed, then P = L and P is a closed invariant equivalence relation in X. Theorem 4 establishes the productivity of the relations Land M, and in Theorem 7, it is shown that {X,T) is coterminous, i.e., P = Q = L= M, iff P = Q.Theorems 9 through 13 describe L, M, P, and Q under various hypotheses such as (X,T) distal; iX,T) minimal and Tabelian; (X,T) regionally mixing; and the like.Theorem 14 is an application of the general theory to obtain a characterization of iX,T) being uniformly equicontinuous, and as such represents a strengthening of a theorem by John D. Baum [2].The author is indebted to Professor W. H. Gottschalk for his invaluable suggestions, mathematical stimulation, and sustained interest.Standing notation.Let(Z,T)be a transformation group where X is always a uniform space.Let sé be the class of all syndetic subsets of T, let J>f be the class or all compact subsets of T, let ^ be the uniformity of X, and for each x e X let jVx be the neighborhood filter of x.Definition 1. (1) The isimply) proximal relation ofiX,T) denoted PiX,T)of PiX) or Px or P, is defined to be the set of all (x,y) e X x X such that if <x e <?/, then there exists t e T such that (xi, yt)

Proximity-based unification theory

This paper is concerned with the topological structure of solvmanifolds i.e. spaces on which solvable Lie groups of transformations act transitively. The special case of nilmanifolds in which the group of transformations is nilpotent has been studied by Malcev [5]. Examples of two dimensional solvmanifolds are the torus, Klein bottle, cylinder, mobius strip. The central results are THEOREM A. Two compact solvmanifolds having the same fundamental group are isomorphic. THEOREM B. Any solvmanifold is covered a finite number of times by the direct product of a compact solvmanifold and a euclidean space; moreover, the covering group is abelian. THEOREM C. If S is an algebraically subgroup (cf. ?9 for definition) of the solvable Lie group G, then G/S is homeomorphic to the direct product of a compact solvmanifold and a euclidean space. Theorem C applies for example, if S is a closed subgroup which is generated by its connected identity-component together with elements s1, s2, -such that all the eigenvalues of Adsi are positive (i = 1, 2, * ). The assertion in Theorem A, applied to nilmanifolds, was proved by Malcev [5]. The situation in that case is simplified by virtue of the fact that two nilpotent groups which operate transitively and effectively on spaces with same fundamental group are isomorphic-a fact which is not true in the case of general solvmanifolds. Accordingly, there is a natural homeomorphism between nilmanifolds with isomorphic fundamental groups but there is no natural homeomorphism in the case of solvmanifolds. In the absence of uniqueness, therefore, proof of the existence of the homeomorphisin in Theorem A is rather elusive. After a number of algebraic maneuvers, proof of Theorem A is reduced to comparing two fibre bundles over homeomorphic toroids-the fibres being the orbits of transformation groups which are isomorphic as abstract topological groups. The main topological lemma asserts the existence of an equivariant homeomorphism between the two bundles. The major algebraic fact which underlies our method is the theorem of ?5 on uniform subgroups of solvable groups. In Theorem B, the covering need not be univalent-as is illustrated by the mobius band. The sufficiency condition in Theorem C yields as a special case a result of Chevalley: the factor space of a solvable Lie group by a connected subgroup is a product of circles and straight lines. Theorem A yields more generally

Let X be a uniform space, let T be a multiplicative topological group, and let T act as a transformation group on X. A subset A of T is said to be (left) syndetic provided that T= AK for some compact subset K of T. T he transformation group T is said to be almost periodic on X provided that if a is an index of X, then there exists a syndetic subset A of T such that xA Cxa for all xEX. If xGX, then the transformation group T is said to be locally almost periodic at x provided that if U is a neighborhood of x, then there exist a neighborhood V of x and a syndetic subset A of T such that VA CU. The transformation group T is said to be locally almost periodic on X in case T is locally almost periodic at every point of X. If xEX, then the transformation group T is said to be locally weakly almost periodic at x provided that if U is a neighborhood of x, then there exist a neighborhood V of x, a syndetic subset A of T, and a compact subset C of T such that y E V implies the existence of a subset B of T for which A CBC and yBC U. It is readily proved that if xcX, then T is locally weakly almost periodic at x if and only if for each neighborhood U of x there exist a neighborhood V of x and a compact subset K of T such that VTC UK. If xEX, then the transformation group T is said to be equicontinuous at x provided that if a is an index of X, then there exists an index e of X such that xftCxta for all tC T. The transformation group T is said to be equicontinuous on X in case T is equicontinuous at every point of X. The transformation group T is said to be uniformly equicontinuous on X provided that if a is an index of X, then there exists an index ,B of X such that xl3tCxta for all xEX and all tcT. It is readily verified that if X is compact, then T is uniformly equicontinuous if and only if T is equicontinuous. The transformation group T is said to be distal on X provided that if x, yEX with x5-zy, then there exists an index a of X such that (xt, yt) Ea for all t E T. We also consider T to be a transformation group acting on X X X in the following manner: if x, yEX and if tET, then (x, y)t is defined to be (xt, yt).

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.

We discuss (linear) geometrical optics and its scalar wave counterpart, Fourier optics, as a natural outgrowth of the group theoretical machinery emanating from the symplectic structure of (optical) phase space, as it derives from the Fermat principle. We do, in fact, quantize geometrical optics. The symplectic structure is preserved by the (symplectic) group of linear canonical transformations. Every element of the symplectic group can be realized as a part of an image system in geometrical optics. On the other hand, one may associate a (nilpotent) Lie algebra to the (symplectic) phase space on which the symplectic group Sp (4, ℜ) acts in the natural way. Looking further on the irreducible unitary representations of the underlying simply connected Lie group, the Heisenberg group, one finds that the symplectic action on the Heisenberg group induces a group of unitary transformations in the space of each (infinite-dimensional) irreducible representation. This group, called the metaplectic group, describes the evolution of the (scalar) wave field (Fourier optics) corresponding to the evolution of rays (geometrical optics), and thereby connects the two views on optics. We put emphasis on the interplay of concepts, and for this reason, we also invoke intuition and terminology from classical and quantum mechanics.

Consider a continuous flow on a locally compact, separable, metric space.If the set of nonperiodic recurrent points is nowhere dense, then there is an open, dense, invariant subset of the phase space which has a Hausdorff orbit space.A separatrix is defined to be a trajectory which is in the closure of the set of trajectories at which the orbit space is not Hausdorff.If the flow is completely unstable, then the set of points which lie on sparatrices is nowhere dense in the phase space.Introduction.Orbit spaces and their topological properties arise in a natural way when the trajectorial structure of a continuous flow is studied, e.g., in [2],[4], and [6].One of the fundamental concepts of trajectorial structure is that of a parallelizable flow.Parallelizable flows have been studied and characterized in [1], [2], and [7, 2.4].If the flow is parallelizable, then the natural mapping of the phase space onto the orbit space is the projection of a fiber bundle.One immediate consequence of parallelizability is that the orbit space is Hausdorff.If a flow on a locally compact, separable, metric space has a Hausdorff orbit space, then there is a natural generalization of the fiber bundle structure for parallelizable flows [6].Hence, flows with Hausdorff orbit spaces are a natural generalization of flows which are parallelizable.Flows with Hausdorff orbit spaces also arise in the study of completely unstable flows [4].Clearly, it is very restive to assume that a flow has a Hausdorff orbit space.In this paper we will show that if a flow on a locally compact, separable, metric space has a nowhere dense set of nonperiodic recurrent points, then there is an open, dense, invariant subset U of the phase space such that the orbit space with respect to U is Hausdorff.A separatrix is a trajectory which is in the closure of the set of trajectories at which the orbit space is not Hausdorff.In the case that the flow is completely unstable, we will show that the set of points which lie on sparatrices is nowhere dense in the phase-space.1. Basic definitions and properties of continuous flow.Throughout this paper X will be a locally compact, separable, metric space and R, R+ will denote the reals and nonnegative reals, respectively.A continuous flow tt on A-is a continuous mapping of X X R onto X

§ i. General principal bundles i. F-structures Let P=(P,~,B,G) be a principal bundle where P and B are topological spaces and @ is a topological group. Let 0: F+G be a continuous homomorphism from a topological group F onto G with kernel K • 0 will be called central, if (i) K is discrete (ii) K is contained in the center of F Thus, in particular, K is abelian. It is easy to check that (ii) follows from (i) if F is connected. Thus every covering projection P: F÷G (F connected) satisfies the conditions above. Throughout this paper we shall assume that p is central. A r-structure on P is a F-principal bundle P=(P,~,B,F) together with a strong bundle map ~: P÷P which is equivariant under the right actions of the structure groups; that is

The structure of the automorphism group of a simple TAI algebra is studied. In particular, we show that \({\frac{\overline{ \rm{Inn}}(A) }{ \overline{\rm{Inn}}_{0} ( A ) }}\) is isomorphic (as a topological group) to an inverse limit of discrete abelian groups for a unital, simple, AH algebra with bounded dimension growth. Consequently, \({\frac{\overline{\rm{Inn}}(A)}{\overline{\rm{Inn}}_{0}(A) }}\) is totally disconnected.Another consequence of our results is the following: Suppose A is the transformation group C*-algebra of a minimal Furstenberg transformation with a unique invariant probability measure. Then the automorphism group of A is an extension of a simple topological group by the discrete group \({\rm{Aut}(\underline{K}(A))_{+,1}}\).

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