A Piecewise Rotation of the Circle, IPR Maps and Their Connection with Translation Surfaces

For given positive integers d j , \(1 \leqslant j \leqslant \forall s\), \(\sum _{j=1}^{s} d_{j}\) even, we construct a piecewise rotation map of the circle with \(\sum _{j=1}^{s} d_{j} \, + \, s\) discontinuous points such that its critical iterates generate translation surfaces with singularity orders d j , \(1 \leqslant j \leqslant s\), and with any Rauzy class associated to this singularity orders. The construction of the piecewise rotation map is combinatorial, on the other hand, the construction of the translation surfaces is based on the idea by Cruz and da Rocha.