### Abstract

For given positive integers d_{j}, 1 ⩽ j⩽ ∀ s, ∑j=1sdj even, we construct a piecewise rotation map of the circle with ∑_{j=1} ^{s}dj+s discontinuous points such that its critical iterates generate translation surfaces with singularity orders d_{j}, 1 ⩽ j⩽ 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.

Original language | English |
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Title of host publication | Lecture Notes in Mathematics |

Publisher | Springer Verlag |

Pages | 393-431 |

Number of pages | 39 |

DOIs | |

Publication status | Published - 2018 Jan 1 |

### Publication series

Name | Lecture Notes in Mathematics |
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Volume | 2213 |

ISSN (Print) | 0075-8434 |

### ASJC Scopus subject areas

- Algebra and Number Theory

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## Cite this

Inoue, K., & Nakada, H. (2018). A Piecewise rotation of the circle, IPR maps and their connection with translation surfaces. In

*Lecture Notes in Mathematics*(pp. 393-431). (Lecture Notes in Mathematics; Vol. 2213). Springer Verlag. https://doi.org/10.1007/978-3-319-74908-2_19