Abstract
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.
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Acknowledgements
The second author was partially supported by JSPS grants No. 16K13766 and JSPS Core-to-core program, “Foundation of a Global Research Cooperative Center in Mathematics focused on Number Theory and Geometry”.
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Inoue, K., Nakada, H. (2018). A Piecewise Rotation of the Circle, IPR Maps and Their Connection with Translation Surfaces. In: Ferenczi, S., Kułaga-Przymus, J., Lemańczyk, M. (eds) Ergodic Theory and Dynamical Systems in their Interactions with Arithmetics and Combinatorics. Lecture Notes in Mathematics, vol 2213. Springer, Cham. https://doi.org/10.1007/978-3-319-74908-2_19
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DOI: https://doi.org/10.1007/978-3-319-74908-2_19
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