Publisher Summary The theory of mathematical billiards can be partitioned into three areas:
convex billiards with smooth boundaries, billiards in polygons (and polyhedra), and …

Publisher Summary Dynamical systems have grown from various roots into a field of great
diversity that interacts with many branches of mathematics as well as with the sciences. This …

This is an updated and expanded version of our earlier survey article [E. Gutkin,“Billiard
dynamics: a survey with the emphasis on open problems,” Regular Chaotic Dyn. 8, 1–13 …

A translation manifold is a manifold whose transition transformations are translations. There
is an important connection between the geometry and arithmetic of translation surfaces and …

We study the topological entropy of a class of transformations with mild singularities: the
generalized polygon exchanges. This class contains, in particular, polygonal billiards. Our …

A billiard ball, ie a point mass, moves inside a polygon Q c JR2 with unit speed along a
straight line until it reaches the boundary 0Q, then instantaneously changes direction …

A polytope exchange transformation is a (discontinuous) map from a polytope to itself that is
a translation wherever it is defined. The 1-dimensional examples, interval exchange …

We investigate a remarkable new planar piecewise isometry whose generating map is a
permutation of four cones. For this system we prove the coexistence of an infinite number of …

We summarize the results of several recent papers, together with a few new results, which
rely on a connection between semi-dispersing billiards and non-regular Riemannian …

{Planar piecewise isometries (PWIs) are iterated mappings of subsets of the plane that
preserve length (and hence angle and area) on each of a number of disjoint regions. They …