Combinatorial and Computational aspects of Tilings

Two-dimensional piecewise isometries provide an inexaustible supply of packings by polygons or ellipses, which are little understood. These systems have zero topological entropy, and hence don't display the elliptic-hyperbolic behaviour of generic smooth symplectic maps. A role analogous to hyperbolic points is played by the so-called pseudo-hyperbolic points, which are the points that recur to the boundary of the atoms.
We develop an arithmetical characterization of these points, for a one-parameter family of piecewise isometries. Using
non-archimedean methods, we prove the non-existence of
unstable cycles, for rational and transcendental values of the parameter. We also discuss how the interplay between elliptic and pseudo-hyperbolic points may be used to develop a bifurcation theory for these systems.