## Abstract

We describe a general method of arithmetic coding of geodesics on the modular surface based on the study of one-dimensional Gauss-like maps associated to a two-parameter family of continued fractions introduced in [Katok and Ugarcovici. Structure of attractors for (a,b)-continued fraction transformations. J. Modern Dynamics 4 (2010), 637-691]. The finite rectangular structure of the attractors of the natural extension maps and the corresponding reduction theory play an essential role. In special cases, when an (a,b)-expansion admits a so-called dual, the coding sequences are obtained by juxtaposition of the boundary expansions of the fixed points, and the set of coding sequences is a countable sofic shift. We also prove that the natural extension maps are Bernoulli shifts and compute the density of the absolutely continuous invariant measure and the measure-theoretic entropy of the one-dimensional map.

Original language | English (US) |
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Pages (from-to) | 739-762 |

Number of pages | 24 |

Journal | Ergodic Theory and Dynamical Systems |

Volume | 32 |

Issue number | 2 |

DOIs | |

State | Published - Apr 2012 |

## All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics