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## О динамике бифуркационных диффеоморфизмов простой дуги

In this paper we consider a class Phi of diffeomorphisms of a closed n -dimensional manifold that are bifurcation points of simple arcs in the space of diffeomorphisms. The authors have studied the asymptotic properties and the embedding structure of invariant manifolds of non-wandering points of such diffeomorphisms.

The results obtained in this paper are related to the Palis–Pugh problem on the existence of an arc withfinitely or countably many bifurcations which joins two Morse–Smale systems on a closed smooth manifoldMn . Newhouse and Peixoto showed that such an arc joining flows exists for any nand, moreover, it is simple. However, there exist isotopic diffeomorphisms which cannot be joined by a simple arc. Forn=1, this is related to the presence of the Poincar´ erotationnumber, and forn=2, to the possible existence of periodic points of different periods and heteroclinic orbits. In this paper, for the dimensionn=3, a new obstruction to the existence of a simple arc is revealed, which is related to the wild embedding of all separatrices of saddle points. Necessary and sufficient conditions for a Morse–Smale diffeomorphism on the 3-sphere without heteroclinic intersections to be joined by a simple arc with a“source-sink”diffeomorphism are also found.

The article argues that the new legitimacy of the Russian regime has a military-mobilization character, in order to maintain it the regime a deeper confrontation with the West and a dismantling of the residual elements of the electoral legitimacy. This shift also requires stronger reliance on the use of force.

The problem of the existence of a simple arc connecting two structurally stable systems on a closed manifold is included in the list of the fifty most important problems of dynamical systems. This problem was solved by S. Newhouse and M. Peixoto for Morse-Smale flows on an arbitrary closed manifold in 1980. As follows from the works of Sh. Matsumoto, P. Blanchard, V. Grines, E. Nozdrinova, O. Pochinka, for the Morse-Smale cascades, obstructions to the existence of such an arc exist on closed manifolds of any dimension. In these works, necessary and sufficient conditions for belonging to the same simple isotopic class for gradient-like diffeomorphisms on a surface or a three-dimensional sphere were found. This article is the next step in this direction. Namely, the author has established that all orientation-reversing diffeomorphisms of a circle are in one component of a simple connection, whereas the simple isotopy class of an orientation-preserving transformation of a circle is completely determined by the rotation number of Poincare.

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.