### Abstract

We describe a family of arithmetic cross-sections for geodesic ow on compact surfaces of constant negative curvature based on the study of generalized Bowen-Series boundary maps associated to cocompact torsion-free Fuchsian groups and their natural extensions, introduced by Katok and Ugarcovici (2017). If the boundary map satisfies the short cycle property, i.e., the forward orbits at each discontinuity point coincide after one step, the natural extension map has a global attractor with finite rectangular structure, and the associated arithmetic cross-section is parametrized by the attractor. This construction allows us to represent the geodesic ow as a special ow over a symbolic system of coding sequences. In the special cases where the "cycle ends" are discontinuity points of the boundary maps, the resulting symbolic system is sofic. Thus we extend and in some ways simplify several results of Adler-Flatto (1991). We also compute the measure-theoretic entropy of the boundary maps.

Original language | English (US) |
---|---|

Pages (from-to) | 167-202 |

Number of pages | 36 |

Journal | Studia Mathematica |

Volume | 246 |

Issue number | 2 |

DOIs | |

State | Published - Jan 1 2019 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

}

*Studia Mathematica*, vol. 246, no. 2, pp. 167-202. https://doi.org/10.4064/sm171020-1-3

**Adler and Flatto revisited : Cross-sections for geodesic ow on compact surfaces of constant negative curvature.** / Abrams, Adam; Katok, Svetlana.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Adler and Flatto revisited

T2 - Cross-sections for geodesic ow on compact surfaces of constant negative curvature

AU - Abrams, Adam

AU - Katok, Svetlana

PY - 2019/1/1

Y1 - 2019/1/1

N2 - We describe a family of arithmetic cross-sections for geodesic ow on compact surfaces of constant negative curvature based on the study of generalized Bowen-Series boundary maps associated to cocompact torsion-free Fuchsian groups and their natural extensions, introduced by Katok and Ugarcovici (2017). If the boundary map satisfies the short cycle property, i.e., the forward orbits at each discontinuity point coincide after one step, the natural extension map has a global attractor with finite rectangular structure, and the associated arithmetic cross-section is parametrized by the attractor. This construction allows us to represent the geodesic ow as a special ow over a symbolic system of coding sequences. In the special cases where the "cycle ends" are discontinuity points of the boundary maps, the resulting symbolic system is sofic. Thus we extend and in some ways simplify several results of Adler-Flatto (1991). We also compute the measure-theoretic entropy of the boundary maps.

AB - We describe a family of arithmetic cross-sections for geodesic ow on compact surfaces of constant negative curvature based on the study of generalized Bowen-Series boundary maps associated to cocompact torsion-free Fuchsian groups and their natural extensions, introduced by Katok and Ugarcovici (2017). If the boundary map satisfies the short cycle property, i.e., the forward orbits at each discontinuity point coincide after one step, the natural extension map has a global attractor with finite rectangular structure, and the associated arithmetic cross-section is parametrized by the attractor. This construction allows us to represent the geodesic ow as a special ow over a symbolic system of coding sequences. In the special cases where the "cycle ends" are discontinuity points of the boundary maps, the resulting symbolic system is sofic. Thus we extend and in some ways simplify several results of Adler-Flatto (1991). We also compute the measure-theoretic entropy of the boundary maps.

UR - http://www.scopus.com/inward/record.url?scp=85060843478&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85060843478&partnerID=8YFLogxK

U2 - 10.4064/sm171020-1-3

DO - 10.4064/sm171020-1-3

M3 - Article

AN - SCOPUS:85060843478

VL - 246

SP - 167

EP - 202

JO - Studia Mathematica

JF - Studia Mathematica

SN - 0039-3223

IS - 2

ER -