We classify when local instability of orbits of closeby points can occur for billiards in two dimensional polygons, for billiards inside three dimensional polyhedra and for geodesic flows on surfaces of three dimensional polyhedra. We sharpen a theorem of Boldrighini, Keane and Marchetti. We show that polygonal and polyhedral billiards have zero topological entropy. We also prove that billiards in polygons are positive expansive when restricted to the set of non-periodic points. The methods used are elementary geometry and symbolic dynamics.

This article demonstrates that in the Lobatchevsky space and on a sphere of arbitrary dimensions, the concept of the mass center of a system of mass points can be correctly defined. Presented are: a uniform geometric construction for defining the mass center; hyperbolic and spheric "lever rules"; the theorem of uniqueness for determining the mass center in these spaces. Among the compact manifolds, only the sphere possesses this property.

Abstract. We show that periodic orbits are dense in the phase space for billiards in polygons for which the angle between each pair of sides is a rational multiple of π.

This paper proves the existence of non-periodic and not everywhere dense billiard trajectories in convex polygons and polyhedrons. For any n ^ 3 there exists a corresponding convex rc-agon (for n = 3 this will be a right triangle with a small acute angle), while in three-dimensional space it will be a prism, the n-agon serving as the base.The results are applied for investigating a mechanical system of two absolutely elastic balls on a segment, and also for proving the existence of an infinite number of periodic trajectories in the given polygons.The exchange transformation of two intervals is used for proving the theorems. An arbitrary exchange transformation of any number of intervals can also be modeled by a billiard trajectory in some convex polygon with many sides.