In his recent papers he defines a new type of piecewise isometry, called Polytope ExchangeTransformation (PET for short) and shows that these maps describe the compactification ofthe outer billiard outside a kite. He describes the first example of a piecewise isometry of the plane with an unboundedorbit. Thismap has known a lot of developments in recent years with the work of Schwartz:, , and. An important class of piecewise isometries is the outer billiard. The first general result has been obtained by Buzzi, proving that every piecewise isometryhas zero entropy, see. Since them,different examples have been examined in order to exhibit different behaviors, see for example. The strict case of the dimension two began ten years ago with the paper of. Remark thatthe case with non oriented interval exchange is more difficult (and called interval exchange withflips). This class of maps has been well studied in dimension one withthe example of the interval exchange maps, see : the map is bijective, equal to the identityoutside a compact interval and the isometries which locally define T are translations. Thusalmost every point of X has an orbit under T and we will study this dynamical system ( X, T )where X \ X is of zero measure. Now considerthe pre-images of the union of the hyperplanes by T : it is a set of zero Lebesgue measure. The piecewise isometry is amap T from X to R n locally defined on each connected set as an isometry of R n. The methods use some cocyclesand a continued fraction algorithm.Ī piecewise isometry in R n is defined in the following way: consider a finite set of hyperplanes,the complement X of their union has several connected components. We prove that a renormalization scheme exists inside this family andcompute the Hausdorff dimension of the discontinuity set. This family is similar to the ones studied byHooper and Schwartz. We introduce a family of piecewise isometries. ∗ Nicolas B´edaride †, Jean-Fran¸cois Bertazzon ‡ ABSTRACT Computation of the Hausdorff dimension ofthe aperiodic set.
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