Let us denote by the Hilbert space of all functions on the measure space Ω, square-integrable with respect to the measure μ.

In the study of the motion of a particle with negligible mass in the gravitational field created by other bodies (for example, the motion of the comet within the Solar system) it is natural to decompose its trajectory into regular and singular parts.

The Legendre manifolds are the maximal integral submanifolds of a contact manifold. The shortest path problem on a manifold with boundary leads to Legendre varieties. We find normal forms of their generic singularities in terms of binary forms invariants theory.

In this paper we study the ergodic properties of the geodesic flows on compact manifolds of non-positive curvature. We prove that the geodesic flow is ergodic and Bernoulli if there exists a geodesic γ such that there is no parallel Jacobi field along γ orthogonal to γ. In particular, this is true if there exists a tangent vector v such that the sectional curvature is strictly negative for all two-planes containing v, or if there exists a tangent vector v such that the second fundamental form of the horosphere determined by v is definite at the support of v.

We consider certain non-invertible maps of the square which are extensions of the quadratic maps of the interval and their small perturbations. We show that several maps of the type possess attractors which are not hyperbolic but have invariant measures similar to Bowen-Ruelle measures for hyperbolic attractors.

We study asymptotic growth of closed geodesies for various Riemannian metrics on a compact manifold which carries a metric of negative sectional curvature. Our approach makes use of both variational and dynamical description of geodesies and can be described as an asymptotic version of length-area method. We also obtain various inequalities between topological and measure-theoretic entropies of the geodesic flows for different metrics on the same manifold. Our method works especially well for any metric conformally equivalent to a metric of constant negative curvature. For a surface with negative Euler characteristics every Riemannian metric has this property due to a classical regularization theorem. This allows us to prove that every metric of non-constant curvature has strictly more close geodesies of length at most T for sufficiently large T then any metric of constant curvature of the same total area. In addition the common value of topological and measure-theoretic entropies for metrics of constant negative curvature with the fixed area separates the values of two entropies for other metrics with the same area.

It is known that for any sequence X1, X2…, of identically distributed independent random matrices with a common distribution μ. the limit

exists with probability 1. We study some conditions under which Λ(μk)→Λ(μ) provided μk → μ in the weak sense.

It is shown that for n ≥ 3 the Lebesgue measure is the unique finitely-additive isometry-invariant measure on the ring of bounded Lebesgue measurable subsets of the n-dimensional Euclidean space.

Consider the billiard ball problem in an open, convex, bounded region of the plane whose boundary is C2 and has at least one point of zero curvature. Then there are trajectories which come arbitrarily close to being positively tangent to the boundary and also come arbitrarily close to being negatively tangent to the boundary.

We give an example of a smooth map of an interval into itself, conjugate to the Feigenbaum map, for which the attracting Cantor set has positive Lebesgue measure.

We consider iterates of absolutely continuous measures concentrated in a neighbourhood of a partially hyperbolic attractor. It is shown that limit points can be measures which have conditional measures of a special form for any partition into subsets of unstable manifolds.

We find very simple examples of C∞-arcs of diffeomorphisms of the two-dimensional torus, preserving the Lebesgue measure and having the following properties: (1) the beginning of an arc is inside the set of Anosov diffeomorphisms; (2) after the bifurcation parameter every diffeomorphism has an elliptic fixed point with the first Birkhoff invariant non-zero (the KAM situation) and an invariant open area with almost everywhere non-zero Lyapunov characteristic exponents, moreover where the diffeomorphism has Bernoulli property; (3) the arc is real-analytic except on two circles (for each value of parameter) which are inside the Bernoulli property area.

We classify up to an isomorphism all factors of the classical horocycle flow on the unit tangent bundle of a surface of constant negative curvature with finite volume.

Sullivan's geometric measure on a geometrically finite hyperbolic manifold is shown to satisfy a mean ergodic theorem on horospheres and through this that the geodesic flow is Bernoulli.

Let M be a compact Riemannian manifold of (variable) negative curvature. Let h be the topological entropy and hμ the measure entropy for the geodesic flow on the unit tangent bundle to M. Estimates for h and hμ in terms of the ‘geometry’ of M are derived. Connections with and applications to other geometric questions are discussed.

It is proved that for a sequence of arbitrarily small piecewise linear perturbations of the twist map, there is a domain with stochastic behaviour (almost hyperbolicity). The measure of this domain has the asymptotics

where A is the magnitude of the perturbation.