Two questions on the topology of compact energy surfaces of natural two degrees of freedom Hamiltonian systems in a magnetic field are discussed. We show that the topology of this 3-manifold (if it is not a unit tangent bundle) is uniquely determined by the Euler characteristic of the accessible region in configuration space. In this class of 3-manifolds for most cases there does not exist a tr...

It is well known that (i) for every irrational number α the Kronecker sequence mα (m = 1, . . . ,M) is equidistributed modulo one in the limit M → ∞, and (ii) closed horocycles of length l become equidistributed in the unit tangent bundle T1M of a hyperbolic surface M of finite area, as l → ∞. In the present paper both equidistribution problems are studied simultaneously: we prove that for any ...

There exists a canonical symplectic structure on the cotangent bundle of a differentiable manifold and similarly a canonical contact structure on the positive projectivization of the cotangent bundle under the scaling R action. Given a Riemannian (or Finsler) metric on the original manifold, this contact structure is contactomorphic to the contact structure determined by the metric on the unit ...

Given a pair (semispray $S$, metric $g$) on a tangent bundle, the family of nonlinear connections $N$ such that $g$ is recurrent with respect to $(S, N)$ with a fixed recurrent factor is determined by using the Obata tensors. In particular, we obtain a characterization for a pair $(N, g)$ to be recurrent as well as for the triple $(S, stackrel{c}{N}, g)$ where $stackrel{c}{N}$ is the canonical ...

Problems related to symmetries and dimensional reduction are common in the mathematical and physical literature, and are intensively studied presently. As a rule, the symmetry group (“reducing group”) and its orbits (“external dimensions”) are compact, and this is essential in models where the volume of the orbits is related to physical quantities. But this case is only a part of the natural pr...

We provide a self-contained, accessible introduction to Ratner’s Equidistribution Theorem in the special case of horocyclic flow on a complete hyperbolic surface of finite area. This equidistribution result was first obtained in the early 1980s by Dani and Smillie [DS84] and later reappeared as an illustrative special case [Rat92] of Ratner’s work [Rat91-Rat94] on the equidistribution of unipot...

at each point z 2 H, making it a model of the hyperbolic plane. Consider the closed compact surface = H= . We aim to show that its isometry group is …nite, and furthermore that we can achieve a sharp upper bound directly proportional to the genus of the surface. Consider, by comparison, the isometries of R=Z, the Euclidean torus T . Any Euclidean translation of R descends to an isometry of T , ...

We give a new simple construction of the maximal entropy invariant measure (the Bowen{Margulis measure) for the geodesic ow on a compact negatively curved manifold. This construction directly connects invariant`conformal density" arising from the Patterson measure on the sphere at innnity of the universal covering space with the maximal entropy measureof the geodesic ow. 0. Introduction Let M b...

In this note we shall study the lifts of stable foliations of pseudo-Anosov diffeomorphism to certain infinite abelian covers. This is motivated, at least in part, by recent progress in understanding the ergodic properties of the analogous horocycle flows on infinite surfaces [3],[13]. Our aim is to show that many of the results from that context hold in this natural and technically simpler set...

We define the category of manifolds with extended tangent bundles, we study their symmetries and we consider the analogue of equivariant cohomology for actions of Lie groups in this category. We show that when the action preserves the splitting of the extended tangent bundle, our definition of extended equivariant cohomology agrees with the twisted equivariant de Rham model of Cartan, and for t...