American Institute of Mathematical Sciences

February  2010, 27(1): 369-382. doi: 10.3934/dcds.2010.27.369

A note on the coding of orbits in certain discontinuous maps

 1 Centro de Matemática da Universidade do Porto, Rua do Campo Alegre, 4619 - 007 PORTO, Portugal

Received  March 2008 Revised  May 2009 Published  February 2010

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 $\mathcal{D}$.
Citation: Miguel Mendes. A note on the coding of orbits in certain discontinuous maps. Discrete & Continuous Dynamical Systems, 2010, 27 (1) : 369-382. doi: 10.3934/dcds.2010.27.369
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