Transverse Equivalence of Complete Conformal Foliations
We study the problem of classification of complete non-Riemannian conformal foliations of codimension q > 2 with respect to transverse equivalence. It is proved that two such foliations are transversally equivalent if and only if their global holonomy groups are conjugate in the group of conformal transformations of the q-dimensional sphere Conf (Sq). Moreover, any countable essential subgroup of the group Conf (Sq) is realized as the global holonomy group of some non-Riemannian conformal foliation of codimension q.
We prove that every complete foliation (M, F) of codimension q > 1 is either Riemannian or a (Conf (S^q), S^q)-foliation. We further prove that if (M, F) is not Riemannian, it has a global attractor which is either a nontrivial minimal set or a closed leaf or a union of two closed leaves. In particular, every proper conformal non-Riemannian foliation (M, F) has a global attractor which is either a closed leaf or a union of two closed leaves, and the space of all non-closed leaves is a connected q-dimensional orbifold. We show that every countable group of conformal transformations of the sphere S^q can be realized as a global holonomy group of complete conformal foliation. Examples of complete conformal foliations with exceptional and exotic minimal sets as global attractors are constructed.
As an application of our previous results we prove theorems of local and global stability of leaves in sense of Ehresmann and Reeb for conformal foliations of codimention $q>2$. It has been shown that for transversally affine foliations the analogous statements on noncompact closed leaves are not valid. We also remind our rusults about local and global stability of compact leaves of foliations with quasi analytical holonomy pseudogroup admitting an Ehresmann connection and corresponding results of other authors.
We investigated conformal foliations $(M,F)$ of codimension $q\geq 3$ and proved a criterion for them to be Riemannian. In particular, the application of this criterion allowed us to proof the existence of an attractor that is a minimal set for each non-Riemannian conformal foliation. Moreover, if foliated manifold is compact then non-Riemannian conformal foliation $(M,F)$ is $(Conf(S^q),S^q)$-foliation with finitely many minimal sets. They are all attractors, and each leaf of the foliation belongs to the basin of at least one of them. The specificity of the proper conformal foliations is indicated. Special attention is given to complete conformal foliations.
This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.
Complete affine foliations (i.e., foliations admitting the affine geometry as the transversal structure) are investigated. The strong transversal equivalence of complete affine foliations is considered, which is a more refined notion than the transverse equivalence of foliations in the sense of Molino. The classification of complete affine foliations with respect to the strong transversal equivalence is reduced to the classification up to conjugacy of countable subgroups of the affine group $Aff(A^q)$. It is shown that each equivalence class contains a two-dimensional suspended foliation on the manifold, which is an Elenberg--MacLane space of type $K(\pi,1)$.
We prove that any compact manifold whose fundamental group contains an abelian normal subgroup of positive rank can be represented as a leaf of a structurally stable suspended foliation on a compact manifold. In this case, the role of a transversal manifold can be played by an arbitrary manifold. We construct examples of structurally stable foliations that have a compact leaf with infinite solvable fundamental group which is not nilpotent. We also distinguish a class of structurally stable foliations each of whose leaves is compact and locally stable in sense of Ehresmann and Reeb.
A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.
Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.
Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.