О связи динамики градиентно-подобного 3-диффеоморфизма со структурой характеристического пространства
In this paper we consider the class GG orientation preserving gradient-like diffeomorphisms ff defined on a smooth oriented closed surface M2. Establishes that for any such pair of diffeomorphisms there is a dual attractor-repeller Af,Rf, which have a topological dimension not greater than 1 and the orbit space in their Supplement Vf (characteristic space) is homeomorphic to the two-dimensional torus. The immediate consequence of this result is, for example, the same period all of separatrices of a saddle of diffeomorphisms f∈G On the possibility of such representation of the dynamics of the system in the form `source-drain" founded a number of classification results for a structurally stable dynamical systems with dabloidami a set consisting of a finite number of orbits of systems of Morse-Smale. For example, for systems in dimension three, there is always a coherent characteristic space associated with the choice of a one-dimensional dual pairs of attractor-repeller. In dimension two this is not true even in the gradient-like case, however, the paper shows that there is a one-dimensional dual pair, the characteristic of the orbit space which is connected.
In the present paper, a solution to the 33rd Palis-Pugh problem for gradient-like diffeomorphisms of a two-dimensional sphere is obtained. It is precisely shown that with respect to the stable isotopic connectedness relation there exists countable many of equivalence classes of such systems.
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.