Adler and Flatto revisited: Cross-sections for geodesic ow on compact surfaces of constant negative curvature

Adam Abrams, Svetlana Katok

Research output: Contribution to journalArticle

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 languageEnglish (US)
Pages (from-to)167-202
Number of pages36
JournalStudia Mathematica
Volume246
Issue number2
DOIs
StatePublished - Jan 1 2019

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Negative Curvature
Geodesic
Cross section
Natural Extension
Discontinuity
Cycle
Fuchsian Group
Torsion-free Group
Global Attractor
Attractor
Simplify
Coding
Orbit
Entropy
Series

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Cite this

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

In: Studia Mathematica, Vol. 246, No. 2, 01.01.2019, p. 167-202.

Research output: Contribution to journalArticle

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