A note on the coding of orbits in certain discontinuous maps

Abstract

We study certain discontinuous maps by means of a coding map definedon a special partition of the phase space which is such that the points ofdiscontinuity of the map, $\mathcal{D}$, all belong to the union of the boundaries of elements in the partition.    For maps acting locally as homeomorphisms in a compact space, we prove that, if the set of points whosetrajectory comes arbitrarily close to the set of discontinuities is closed and notthe full space then all points not in that set are rationally coded, i.e.,their codings eventually settle on a repeated block of symbols.   In particular, for piecewise isometries, which are discontinuousmaps acting locally as isometries, we give a topological description of theequivalence classes of the coding map in terms of the connected componentsgenerated by the closure of the preimages of $\mathcal{D}$.