Table of contents

For flows on an orientable closed surface of larger genus (that is, of genus ) a special geodesic distribution (the geodesic framework of the flow) is constructed that consists of geodesics with the same asymptotic directions as the trajectories of the flow and that is a complete topological invariant of the irrational flows on such surfaces. The problem of the dependence of the geodesic framework on a perturbation of the flow (or on the parameter of a family of flows) is considered. It is shown that an irreducible elementary irrational geodesic framework of a flow depends continuously on the perturbation of the flow (which is analogous to the continuous dependence of an irrational Poincare rotation number on a perturbation of a flow).

We study the space of Radon probability measures on a metric space and its subspaces , and of continuous measures, discrete measures, and finitely supported measures, respectively. It is proved that for any completely metrizable space , the space is homeomorphic to a Hilbert space. A topological classification is obtained for the pairs , and , where is a metric compactum, an everywhere dense Borel subset of , an everywhere dense -set of , and an everywhere uncountable everywhere dense Borel subset of of sufficiently high Borel class. Conditions on the pair are found that are necessary and sufficient for the pair to be homeomorphic to .

Let be the fundamental group of a compact non-orientable surface of genus  and let be an algebraically closed field of characteristic 0. The structure of the representation varieties , of into and and of the character varieties is described; namely, the number of their irreducible components and their dimensions are determined and their birational properties are investigated. It is proved, in particular, that all the irreducible components of are -rational varieties.

The boundedness of the Hardy operator and the Hardy-Littlewood operator are established, respectively, in and the space BMO of functions of bounded mean oscillation on the real axis . Here the space is isomorphic to the Hardy space of single-valued analytic functions in the upper half-plane satisfying condition (0.3), the Hardy-Littlewood operator is defined in by equality (0.2), and the Hardy operator is defined in by equality (0.1) and its value is continued to as an even (odd) function if the function is even (odd). For an arbitrary function one sets , where is the even and is the odd component of .

The problem of geodesic lines on a two-dimensional torus is considered. One-parameter symmetry groups in the four-dimensional phase space that are generated by vector fields commuting with the initial Hamiltonian vector field are studied. As proved by Kozlov and Bolotin, a geodesic flow on a two-dimensional torus admitting a non-trivial infinitesimal symmetry of degree n has a many-valued integral that is a polynomial of degree at most n in the momentum variables. Kozlov and the present author proved earlier that first- and second-order infinitesimal symmetries are related to hidden cyclic coordinates and separated variables. In the present paper the structure of polynomial infinitesimal symmetries of degree at most four is described under the assumption that these symmetry fields are non-Hamiltonian.

The eigenvalue problem for the Sturm-Liouville operator on the closed interval [0,1] with potential depending on the spectral parameter and with zero Dirichlet boundary conditions is considered first. It is proved under certain assumptions about the potential that if a system of eigenfunctions of this problem contains a unique function with n zeros in the interval (0,1) for each non-negative integer n, then it is complete in the space L₂(0,1) if and only if the functions in this system are linearly independent in L₂(0,1). Next, this result is used in the study of the spectral problem for a certain non-linear operator of Sturm-Liouville type. The completeness in L₂(0,1) of the corresponding eigenfunctions is proved.

In this paper we obtain a criterion for the continuous and smooth orbital equivalence of integrable Hamiltonian systems with degrees of freedom in the vicinity of compact elliptic orbits. Moreover, we construct a complete orbital invariant for a non-degenerate integrable Hamiltonian system with two degrees of freedom in a neighbourhood of an elliptic singular point, and propose a rule from which to compute this orbital invariant. The orbital invariant is computed for integrable Lagrange systems in rigid body dynamics. In this way we find an explicit decomposition of all Lagrange systems into classes of orbitally equivalent ones in the vicinity of equilibria.