We apply methods developed for two-dimensional piecewise isometries to the study of renormalizable interval exchange transformations over an algebraic number field QðÞ, which lead to dynamics on lattices. We consider the Z-module M generated by the translations of the map. On it, we define an infinite family of discrete vector fields, representing the action of the map over the cosets QðÞ=M, which together form an invariant partition of the field QðÞ. We define a recursive symbolic dynamics, with the property that the eventually periodic sequences coincide with the field elements. We apply this approach to the study of a model introduced by Arnoux and Yoccoz, for which À1 is a cubic Pisot number. We show that all cosets of M decompose in a highly non-trivial manner into the union of finitely many orbits.

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 construct a rational piecewise rotation system with three atoms for which the return time to one of the atoms is unbounded. We show that the return map gives rise to a self-similar structure of induced atoms. The constructions are based on the angle of rotation π/7. Moreover, we construct a continuous class of examples with an infinite number of periodic cells. These periodic cells alternate between two atoms and they form a selfsimilar structure. Our investigation here may be viewed as generalizations of results obtained by Boshernitzan and Caroll, as well as Adler, Kitchens and Tresser, Kahng, Lowenstein and others. The main tools in the investigation are algebraic computations in a cyclotomic field determined by fourteenth roots of unity.

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In this paper, we study a class of piecewise rotations on the square. While few theoretical results are known about them, we numerically compute box-counting dimensions, correlation dimensions and complexity of the symbolic language produced by the system. Our results seem to confirm a conjecture that the fractal dimension of the exceptional set is two, as well as indicate that the dynamics on it is not ergodic. We also explore a relationship between the piecewise rotations and discretized rotations on lattices Z 2n . © 2003 American Institute of Physics. ͓DOI: 10.1063/1.1572411͔It is well known that simply defined dynamical systems can exhibit very complex behavior. In this paper we will consider a family of piecewise isometries, namely of discontinuous rotations on the two-torus. These systems are not ergodic; their various invariant sets display a rich and complicated geometry and support dynamics with nontrivial statistical properties. Most of the characteristics of these systems are not completely understood. Basic questions such as the distribution of periodic points and the coding of the trajectories on the closure of the discontinuity lines, are still open. We will present a careful numerical investigation of three features of these maps. First, we compute the fractal dimension of the exceptional set "i.e., the complement of the collection of ''periodic islands''…. There is evidence that this exceptional set may have a positive Lebesgue measure. Second, we investigate the complexity of the orbits on the exceptional set; we found several "piecewise… polynomial behaviors of the complexity function, some referring to a substitution system. Third, we show a correspondence between discontinuous rotations of the torus and discretized rotations on a lattice. This correspondence may help in explaining the differences in the growth of the complexity function.