A note on the coding of orbits in certain discontinuous maps

We study certain discontinuous maps by means of a coding map defined on a special partition of the phase space which is such that the points of discontinuity 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 whose trajectory comes arbitrarily close to the set of discontinuities is closed and not the 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 discontinuous maps acting locally as isometries, we give a topological description of the equivalence classes of the coding map in terms of the connected components generated by the closure of the preimages of $\mathcal{D}$.