## Abstract

. In this paper we give a geometric interpretation of the renormalization algorithm and of the continued fraction map that we introduced in our previous paper (Beyond Sturmian Sequences: coding linear trajectories in the regular octagon, Proceedings London Mathematical Society)
to give a characterization of symbolic sequences for linear ﬂows in the regular
octagon. We interpret this algorithm as renormalization on the Teichm¨uller
disk of the octagon and explain the relation with Teichm¨uller geodesic ﬂow.
This connection is analogous to the classical relation between Sturmian sequences, continued fractions and geodesic ﬂow on the modular surface. We
use this connection to construct the natural extension and the invariant measure for the continued fraction map. We also deﬁne an acceleration of the
continued fraction map which has a ﬁnite invariant measure.

Translated title of the contribution | Geodesic flow on the Teichm\"uller disk of the regular octagon, cutting sequences and octagon continued fractions maps |
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Original language | English |

Title of host publication | Dynamical Numbers: Interplay between Dynamical Systems and Number Theory, Contemporary Mathematics Series |

Editors | S. Kolyada, Y. Manin, M. Moller, P. Moree, T. Ward |

Publisher | American Mathematical Society |

Pages | 29 - 65 |

Number of pages | 35 |

Volume | 532 |

Publication status | Published - 2010 |