Proximity relations in transformation groups

Introduction. In this paper, we investigate the proximal relation and related notions in coset transformation groups.In §1, we show that many of the interesting properties of the proximal relation can be characterized in terms of a group subset that arises naturally from the generating subgroup.A general criterion for distal in terms of subnormality of the generating subgroup is shown in §2.Proofs are given for [9, Remark (2.1) and Theorem (2.2)], establishing that coset transformation groups of nilpotent groups are distal.§3 is concerned with distal in relation to certain coset transformation groups that arise from solvable Lie groups.Finally, we establish in §4 some results concerning the existence of subgroups which yield either point-transitive or minimal distal actions on the phase space.These results are strengthened when applied to certain discrete flows and examined in Lie groups.Proofs for the rest of the results of [9] are given in this section.Let G be a topological group and H a subgroup of G. Consider the space H\G={Hg | ge G} of right cosets of G.The coset transformation group of G induced by H is defined to be the point-to-point transitive transformation group (H\G, G) with action (Hfi g) -> Hfg.The fact that (H\G, G) is indeed a transformation group under this action follows from [6, Definition 1.39].If H is syndetic in G, H\G is compact uniformizable, whence H\G has a unique compatible uniformity.Thus, the proximal relation of (H\G, G) is well defined and will be denoted by P(H\G).Throughout this paper, G will denote a topological group with identity e and 77 will denote a syndetic subgroup.If K and L are subgroups of G, we denote that K is normal in L by K <] L. If K is a subgroup of G, then K* will denote the closure of K.The additive groups of integers and reals will be denoted by Z and R respectively.The neighborhood filter of e will be denoted by r¡e.For other definitions, see [9].Finally, the author would like to thank Professor W. Gottschalk for his stimulating comments concerning the results in this paper.2. The proximal relation.When the phase space is not necessarily uniform, we can still introduce a notion of proximal.

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).

Higher order regionally proximal equivalence relations for general minimal group actions

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)

The purpose of this paper is to give a necessary and sufficient condition in order that the S limit set of a point consists of exactly one periodic orbit which is both S-Lyapunov stable and S-asymptotically stable. The general reference for most terms used in this paper is [2]. In this paper we assume (X, T, r) is a transformation group. X will always be assumed to be a uniform space where the Hausdorff topology in X is induced by the uniformity. T will be assumed to be generative, that is, T is isomorphic to Rm X I X C where R is the additive group of real numbers, I is the additive group of integers, m and n are nonnegative integers, and C is a compact abelian group [4]. S will denote a closed replete semigroup in T. The S-limit set of yEX, Sy, is defined bySv = n te,s Cl (ytS) [1 ]. DEFINITION 1. The orbit of a point yEX is said to be uniformly S-Lyapunov stable with respect to BCX provided that for each index a of X there is an index ,3 of X such that if weEBnyt'13 for some t' E T then wt Eyt'ta for all t E S.

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

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.

First, we discuss the behavior of boundedness in uniform spaces with respect to subspaces, projective limits, and suprema in relation to precompactness. A special uniformly isomorphic embedding of un arbitrary uniform space in a bounded uniform space is presented and examined in 2.6. Hejcman’s characterization (by B-conservativity) of uniform spaces in which boundedness can be tested by a single pseudometric is proved in a new way, see 3.13, using a version 3.1 of the metrization lemma. We comment briefly on boundedness in topological vector spaces. In topological groups we investigate a hierarchy of partly new notions of boundedness, strongly interrelated among themselves, and exhibit various situations in which certain of these notions coincide. Boundedness respecting subspaces of a uniform space prove useful. Many examples illustrate and complement the general theory, see, e.g., Example 6.4.

In Chapters 8–10 we study the general theory of symmetry in science via the conceptual approach. That approach is the best for understanding the concepts involved in symmetry and the significance of symmetry in science. But the application of symmetry considerations in science, and especially quantitative applications, require a general symmetry formalism. That formalism is developed in the present chapter and in Chapters 5–7 following. Mathematical concepts presented in the preceding two chapters are used extensively. Our road to symmetry starts with a state space of a system and proceeds through transformation, transformation group, equivalence relation for a state space, symmetry transformation, and symmetry group. The quantification of symmetry is discyssed, and a special section is devoted to symmetry of quantum systems.

Chaos theory can be used for quantitative dynamics of uncertainty and finding order in its perturbances. Chaos theory is largely a colloquial notion and is referred to as an analysis of non-linear dynamical systems. The studies devoted to linear dynamical systems and theories of complexity involve studies of turbulence, or, more precisely, the transfer from stability to turbulence. The dynamical system by its nature does not follow long-term forecasts. There are two reasons for the unpredictability. Dynamical systems involve both feedback and critical levels. To paraphrase, one can say that a dynamical system forms a system of non-linear feedback. The most important properties of the system include: sensitivity to the change of the initial conditions, occurrence of critical points and a fractal dimension. Non-linear dynamical systems tend to have more than one solution. The case often is that the number of solutions is huge or even infinite. A visual representation of the data forms a finite space called the phase space of the system. The number of dimensions in the space is determined by the number of variable presents in the system. If there are two or three variables, it is possible to visually examine the data. In case of a larger number of dimensions, the data are examined with the use of mathematical data. Another term used in non-linear dynamical systems is the attractor. This is an area of a equilibrium of non-linear system within a time series. A system which aims to achieve an equilibrium in the form of a single value has a point attractor. Besides, there are phase attractors, which form periodic cycles or orbits in space forming a limit cycle. An attractor, in which none of the points in the space overlaps and whose orbits do not cross but both originate in the same area of phase space, is called strange attractor. Such attractors in contrast to point attractors are non-periodic and predominantly have a fractal dimension [10,11].

We apply the techniques of canonical transforms to equations of the type \[\left[ {A( t )\mathbb{P}^2 + B( t )\{ \mathbb{P}\mathbb{Q} + \mathbb{Q}\mathbb{P} \} + C( t )\mathbb{Q}^2 + D( t )\mathbb{Q} + E( t )\mathbb{P} + F( t )\mathbb{1}} \right]\psi ( q,t ) = - i\partial _t \psi ( q,t ),\]where $\mathbb{Q}$ and $\mathbb{P}$ are the quantum position and momentum operators. The time-dependent parameters of the ${\text{W}} \wedge {\text{SL}}( {{\text{2,}}R} )$ evolution operator are found through linear differential equations. In terms of these we give explicitly the Green’s function, all separating coordinates and similarity solutions of the equation. We analyze the behavior of Gaussian and coherent-state initial conditions in closed form and present a new interpretation of all the Lewis-Riesenfeld constants of motion.

The lifetime distribution of energized molecules is examined within the framework of classical description of the molecular dynamics and in the connection with model theories of unimolecular reactions. If the reacting molecule represents an ergodic system, the lifetime distribution function of the molecules represented immediately after activation by the points in the phase space region corresponding to the internal energy ϵ to ϵ + δϵ (RRKM model) may be introduced; in a non‐ergodic case, the distribution function may be defined in that part of this phase space region, the point of which satisfy a supplementary condition of decomposition (Slater's model). It is shown that allowing a rapid intramolecular redistribution of internal energy of the molecule in the originally harmonic model, we obtain lifetime densities approximating the exponential (random) density. Some general features of the lifetime distribution in an ensemble of molecules with a uniform phase space density after activation are deduced from the qualitative analysis of the phase space trajectories.

Employees blood pressure - in control?: Dennis M. Shepard, S.C. Department of Health and Environmental Control, Columbia, S.C.; David B. McCallum

We construct a 4-dimensional phase space using the orbit space by S1-action in hyperbolic 3-space and the Lagrangian by the Hsiang-Lawson metric. We then consider certain integrable systems as Hamiltonian systems on the phase space, and we give a Haantjes operator on the 4-dimensional phase space and construct a concrete symplectic-Haantjes manifold. In particular, we construct this manifold using a rewritten conditional formulation of the definition formula given by Tempesta and Tondo.

In this paper, we present a graphical tool running on a PC for studies of integrable systems of one degree of freedom with several parameters, in particular for perturbed Hamiltonian systems with symmetries whose orbit space is the two-dimensional sphere S2 after they are reduced by normalization. The pioneering work was performed by Coffey et al. [1990], who developed a graphical method on a Connection Machine in order to paint the Hamiltonian function ℋ* over phase space, instead of numerical integrations of the corresponding differential equations. We have pursued their idea and the procedure proposed here renders pictures at the rate of one picture per minute. For research or teaching activities, the above mentioned method may be used in two directions: (a) in the stability analysis of equilibria and their bifurcations in parameter space; (b) for the specification of initial conditions and the corresponding qualitative long term global dynamics, in order to make proper choices for numerical studies of the full problem. As examples of how to interact with the program, and of the different role played by physical and dynamical parameters, we consider the dynamics of the Laplace–Runge–Lenz vector in two perturbed Keplerian systems of recent interest: the generalized Van der Waals effect in the hydrogen atom, and Harrington’s model for triple stellar systems.