de Mathématiques de Luminy
Bressaud Xavier, Poggiaspalla Guillaume.
A classification of bijective polygonal piecewise isometries.
We aim to give a classification of euclidian bijective polygonal piecewise isometries with a finite number of compact polygonal atoms. We rely on a specific type of triangulation process which enables us to describe a notion of combinatorial type similar to its one-dimensional counterpart for interval exchange maps. Moreover, it is possible to handle all the possible piecewise isometries, given two combinatorial types. We show that most of the examples treated in the literature of piecewise isometries can be retrieved by systematic computations. We also exhibit a new class of maps, the piecewise similarities, which fit nicely in this framework and whose behaviour is shown to be highly non-trivial.