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

We present several examples of piecewise isometric systems that give rise to complex
structures of their coding partitions. We also list and comment on current open questions in …

We discuss the dynamics of a class of non-ergodic piecewise affine maps of the torus. These
maps exhibit highly complex and little understood behavior. We present computer graphics …

Let X be a region of ${\rm\bf R}\sp {N} $ and ${\cal P}=\{P\sb0,\..., P\sb {r-1}\} $ a finite $(r> 1)
$ partition of X such that each $ P\sb {i} $ has positive Lebesgue measure. A map $ T:\X\to X …

We show that piecewise isometries, ie non-necessarily invertible maps defined on a finite
union of polytopes and coinciding with an isometry on the interior of each polytope, have …

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 …

We investigate the dynamics of systems generalizing interval exchanges to planar
mappings. Unlike interval exchanges and translations, our mappings, despite the lack of …

We study two piecewise affine maps on convex polygons, locally conjugate to a rotation by a
multiple of π/7. We obtain a finite-order recursive tiling of the phase space by return map sub …

Piecewise rotations are natural generalizations of interval exchange maps. They appear
naturally in the theory of digital filters, Hamiltonian systems and polygonal dual billiards. We …