Renormalization in Piecewise Isometries

Interval exchange transformations (IET) are bijective piecewise translations of an interval divided into a finite partition of subintervals. Piecewise isometries (PWIs) are generalizations of IETs to higher dimension where a region is split into a number of convex sets and these are rearranged using isometries. Although PWIs are higher dimensional generalizations of IETs, their generic dynamical properties seem to be quite different. In this thesis we consider embeddings of IETs into PWIs in order to understand their similarities and differences. We investigate translated cone exchange transformations, a new family of piecewise isometries and renormalize its first return map to a subset of its partition. As a consequence we show that the existence of an embedding of an interval exchange transformation into a map of this family implies the existence of infinitely many bounded invariant sets. We also prove the existence of infinitely many periodic islands, accumulating on the real line, as well as non-ergodicity of our family of maps close to the origin. We derive some necessary conditions for existence of embeddings using combinatorial, topological and measure theoretic properties of IETs. In particular, we prove that continuous embeddings of minimal 2-IETs into orientation preserving PWIs are necessarily trivial and that any 3-PWI has at most one non-trivially continuously embedded minimal 3-IET with the same underlying permutation. Furthermore, we introduce a family of 4-PWIs with apparent abundance of invariant nonsmooth fractal curves supporting IETs, that limit to a trivial embedding of an IET. Finally, we prove that almost every interval exchange transformation, with an associated translation surface of genus $g\geq 2$, can be non-trivially and isometrically embedded into a family of piecewise isometries. In particular, this proves the existence of invariant curves for piecewise isometries, reminiscent of KAM curves for area preserving maps, which are not unions of circle arcs or line segments