Dynamical Systems whose Orbit Spaces are Nearly Hausdorff

For a given Hilbert space $$\mathcal H$$ , consider the space of self-adjoint projections $$\mathcal P(\mathcal H)$$ . In this paper we study the differentiable structure of a canonical sphere bundle over $$\mathcal P(\mathcal H)$$ given by $$\begin{aligned} \mathcal R=\{\, (P,f)\in \mathcal P(\mathcal H)\times \mathcal H \, : \, Pf=f , \, \Vert f\Vert =1\, \}. \end{aligned}$$ We establish the smooth action on $$\mathcal R$$ of the group of unitary operators of $$\mathcal H$$ , and it thereby turns out that the connected components of $$\mathcal R$$ are homogeneous spaces. Then we study the metric structure of $$\mathcal R$$ by endowing it first with the uniform quotient metric, which is a Finsler metric, and we establish minimality results for the geodesics. These are given by certain one-parameter groups of unitary operators, pushed into $$\mathcal R$$ by the natural action of the unitary group. Then we study the restricted bundle $$\mathcal R_2^+$$ given by considering only the projections in the restricted Grassmannian, locally modeled by Hilbert–Schmidt operators. Therefore we endow $$\mathcal R_2^+$$ with a natural Riemannian metric that can be obtained by declaring that the action of the group is a Riemannian submersion. We study the Levi–Civita connection of this metric and establish a Hopf–Rinow theorem for $$\mathcal R_2^+$$ , again obtaining a characterization of the geodesics as the image of certain one-parameter groups with special speeds.

Consider the set źnw0$\chi ^{0}_{\text {nw}}$ of non-wandering continuous flows on a closed surface M. Then we show that such a flow can be approximated by a non-wandering flow v such that the complement MźPer(v) of the set of periodic points is the union of finitely many centers and finitely many homoclinic saddle connections. Using the approximation, the following are equivalent for a continuous non-wandering flow v on a closed connected surface M: (1) the non-wandering flow v is topologically stable in źnw0$\chi ^{0}_{\text {nw}}$; (2) the orbit space M/v is homeomorphic to a closed interval; (3) the closed connected surface M is not homeomorphic to a torus but consists of periodic orbits and at most two centers. Moreover, we show that a closed connected surface has a topologically stable continuous non-wandering flow in źnw0$\chi ^{0}_{\text {nw}}$ if and only if the surface is homeomorphic to either the sphere S2$\mathbb {S}^{2}$, the projective plane ź2$\mathbb {P}^{2}$, or the Klein bottle K2$\mathbb {K}^{2}$.

We formalize and study several competing notions of versality for an action of a linear algebraic group on an algebraic varietyXX. Our main result is that these notions of versality are equivalent to various statements concerning rational points on twisted forms ofXX(existence of rational points, existence of a dense set of rational points, etc.). We give applications of this equivalence in both directions to study versality of group actions and rational points on algebraic varieties. We obtain similar results onpp-versality for a prime integerpp. An appendix, containing a letter from J.-P. Serre, puts the notion of versality in a historical perspective.

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].

Si l'action α du groupe G est monotone, alors la symetrie est incluse dans le comportement non chaotique

The main topic of discussion in this survey is the geometric and metric properties of geodesic and horocycle flows on surfaces of constant curvature and their multidimensional analogues, the so-called G-induced actions (that is, restrictions to a suitable subgroup F of the natural action of G on a homogeneous space of it). One usually takes as F one-parameter or cyclic subgroups of G and the horospherical subgroups corresponding to them (the details are given in Sect. 1). However, in the course of the exposition of our main topic we sometimes mention results relating to the case when F is an n-parameter subgroup of G or a lattice. This class of dynamical systems consists precisely of group actions admitting an extension to a transitive action of a Lie group. We shall not be considering in our survey (or, at any rate, not systematically) the following themes which are close to our main subject matter: Topological dynamics and metric properties of general group actions (in particular, the work of Zimmer on semisimple group actions (Zimmer 1984); Group automorphisms; Affine transformations of homogeneous spaces; G-induced flows on spaces with an infinite invariant measure. The papers (Hopf 1939), (Sullivan 1981) and the recent book (Nicholls 1989) are devoted to the connections between G-induced flows (and also geodesic flows) and the theory of discrete groups of motions of a hyperbolic space; The actions of subgroups on the double cosets related to geodesic (for n > 2) and horocycle (for n > 3) flows on an n-dimensional manifold of constant negative curvature (see (Gel’fand and Fomin 1952) and, in particular, (Mautner 1957)). Flows of this type in somewhat more general form differing from geodesic and horocycle flows have not been investigated.

A vector-fiber bundle structure of the reaction path Hamiltonian, which has been introduced by Miller, Handy and Adams, is explored with respect to molecular vibrations orthogonal to the reaction path. The symmetry of the fiber bundle is characterized by the real orthogonal group O(3N- 7) for the dynamical system with N atoms. Under the action of group O(3N- 7). the kinetic energy of the reaction path Hamiltonian is left invariant. Furthermore , the invariant behaviour of the Hamiltonian vector fields is investigated.

(2008). On Ordering the Natural Numbers or The Sharkovski Theorem. The American Mathematical Monthly: Vol. 115, No. 2, pp. 159-165.

We examine the semicrossed products of a semigroup action by $*$-endomorphisms on a C*-algebra, or more generally of an action on an arbitrary operator algebra by completely contractive endomorphisms. The choice of allowable representations affects the corresponding universal algebra. We seek quite general conditions which will allow us to show that the C*-envelope of the semicrossed product is (a full corner of) a crossed product of an auxiliary C*-algebra by a group action. Our analysis concerns a case-by-case dilation theory on covariant pairs. In the process we determine the C*-envelope for various semicrossed products of (possibly nonselfadjoint) operator algebras by spanning cones and lattice-ordered abelian semigroups. In particular, we show that the C*-envelope of the semicrossed product of C*-dynamical systems by doubly commuting representations of $\mathbb{Z}^n_+$ (by generally non-injective endomorphisms) is the full corner of a C*-crossed product. In consequence we connect the ideal structure of C*-covers to properties of the actions. In particular, when the system is classical, we show that the C*-envelope is simple if and only if the action is injective and minimal. The dilation methods that we use may be applied to non-abelian semigroups. We identify the C*-envelope for actions of the free semigroup $\mathbb{F}_+^n$ by automorphisms in a concrete way, and for injective systems in a more abstract manner. We also deal with C*-dynamical systems over Ore semigroups when the appropriate covariance relation is considered.

Let \Lambda be a finite abelian group. A dynamical system with transformation group \Lambda is a triple (A,\Lambda,\alpha) , consisting of a unital locally convex algebra A , the finite abelian group \Lambda and a group homomorphism \alpha:\Lambda\rightarrow\operatorname{Aut}(A) , which induces an action of \Lambda on A . In this paper we present a new, geometrically oriented approach to the noncommutative geometry of principal bundles with finite abelian structure group based on such dynamical systems.

Essential results about actions of compact Lie groups on connected manifolds are generalized to proper actions of arbitrary groups on connected cohomology manifolds. Slices are replaced by certain fiber bundle structures on orbit neighborhoods. The group dimension is shown to be effectively finite. The orbits of maximal dimension form a dense open connected subset. If some orbit has codimension at most $2$, then the group is effectively a Lie group.

In this paper, the T -orbital ultrametric spaces are introduced and a fixed point theorem for sequentially bounded mappings is given. Our main result extends some known theorems for nonexpansive mappings. Examples are given to support our work.

Let $U$ be a unipotent group which is graded in the sense that it has an extension $H$ by the multiplicative group of the complex numbers such that all the weights of the adjoint action on the Lie algebra of $U$ are strictly positive. We study embeddings of $H$ in a general linear group $G$ which possess Grosshans-like properties. More precisely, suppose $H$ acts on a projective variety $X$ and its action extends to an action of $G$ which is linear with respect to an ample line bundle on $X$. Then, provided that we are willing to twist the linearization of the action of $H$ by a suitable (rational) character of $H$, we find that the $H$-invariants form a finitely generated algebra and hence define a projective variety $X/\!/H$; moreover, the natural morphism from the semistable locus in $X$ to $X/\!/H$ is surjective, and semistable points in $X$ are identified in $X/\!/H$ if and only if the closures of their $H$-orbits meet in the semistable locus. A similar result applies when we replace $X$ by its product with the projective line; this gives us a projective completion of a geometric quotient of a $U$-invariant open subset of $X$ by the action of the unipotent group $U$.

We establish several classification results for compact extensions of tracial W^{*} -dynamical systems and for relatively independent joinings thereof for actions of arbitrary discrete groups. We use these results to answer a question of Austin, Eisner, and Tao and some questions raised by Duvenhage and King. Moreover, combining our results with an earlier classification of weakly mixing extensions by Popa, we can derive non-commutative Furstenberg–Zimmer type dichotomies on the L^{2} -level. Although in general an adequate generalization of the Furstenberg–Zimmer tower of intermediate compact extensions does not seem possible in the von Neumann algebraic framework, we show that there always exists a non-commutative analogue of the finer Host–Kra–Ziegler tower for any ergodic action of a countable abelian group.

In this paper, we discuss sections of higher homotopy groupoid bundles those yield by group actions. We study some basic properties related to the left-invariant (G-invariant) of such sections and also present some conditions for such sections to be left-invariant (G-invariant). We concentrate a kind of group action on the Core higher homotopy groupoid bundle of certain topological spaces which undergo a regular topological group action. Interestingly, each element of orbit space yields continuous sections of the Core higher homotopy groupoid bundle having beautiful algebraic properties. Moreover, each of those continuous sections behaves nicely as a left-invariant section with respect to topological group action by G. Since higher homotopy groups are abelian, the set of such sections forms always an abelian group. Moreover, each element of the group is a left-invariant section, therefore, it is convenient to call them commutative group of left-invariant sections of the Core higher homotopy groupoid bundle. Further, we show this group is a topological invariant and study some basic properties of it. At the end of the paper, we give interrelations between the retraction map and this group.