Luchezar Stojanov

An estimate from above of the number of periodic orbits for semi-dispersed billiards

For a large class of semi-dispersed billiards an exponential estimate from above is found for the number of periodic points of the billiard ball map.

Complete minimal and totally minimal groups

Abstract First we construct complete totally minimal topological groups which are not locally compact. Extending and generalizing a construction of S. Dierolf and U. Schwanengel (using semi-direct products) we find then a class A of topological groups such that all products of elements of A are minimal topological groups. This gives further examples of complete minimal topological groups being not locally compact and partial answers to a question posed by A.V. Arhangelskiǐ.

Periods of multiple reflecting geodesics and inverse spectral results

where AD is a self-adjoint operator in L2(Q) corresponding to the laplacian - A with Dirichlet boundary condition on aQ. The singularities of aD(t) are connected with the periods of all periodic generalized geodesics in Q, including those lying entirely on ai and related to the induced Riemannian metric on aQ. The generalized geodesics are obtained as projections on Q of the generalized periodic bicharacteristics of the wave operator D =

Generic properties of periodic reflecting rays

It is shown that for generic domains D in n , n ≥ 2, every periodic billiard trajectory in D passes only once through each of its reflection points, and any two different periodic billiard trajectories in D have no common reflection point.

Periodic geodesics of generic nonconvex domains in

Here £Q is the union of all generalized periodic geodesies 7 in Ü, including those lying entirely on dfi, and T^ is the period (length) of 7 (see [1]). Generalized geodesies are projections on Ü of the generalized bicharacteristics of d\ — A, introduced by Melrose and Sjöstrand [6]. We have proved in [8, 9] that for generic strictly convex domains in R the relation (2) becomes an equality and the spectrum of (1) determines the lengths of all periodic geodesies (see [5] for related results). The purpose of this announcement is to prove the same result for generic nonconvex domains in R .