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Symbolic dynamics for the modular surface and beyond


Authors: Svetlana Katok and Ilie Ugarcovici
Journal: Bull. Amer. Math. Soc. 44 (2007), 87-132
MSC (2000): Primary 37D40, 37B40, 20H05
DOI: https://doi.org/10.1090/S0273-0979-06-01115-3
Published electronically: October 2, 2006
MathSciNet review: 2265011
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Abstract: In this expository article we describe the two main methods of representing geodesics on surfaces of constant negative curvature by symbolic sequences and their development. A geometric method stems from a 1898 work of J. Hadamard and was developed by M. Morse in the 1920s. It consists of recording the successive sides of a given fundamental region cut by the geodesic and may be applied to all finitely generated Fuchsian groups. Another method, of arithmetic nature, uses continued fraction expansions of the end points of the geodesic at infinity and is even older--it comes from the Gauss reduction theory. Introduced to dynamics by E. Artin in a 1924 paper, this method was used to exhibit dense geodesics on the modular surface. For 80 years these classical works have provided inspiration for mathematicians and a testing ground for new methods in dynamics, geometry and combinatorial group theory. We present some of the ideas, results (old and recent), and interpretations that illustrate the multiple facets of the subject.


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Additional Information

Svetlana Katok
Affiliation: Department of Mathematics, The Pennsylvania State University, University Park, Pennsylvania 16802
Email: katok_s@math.psu.edu

Ilie Ugarcovici
Affiliation: Department of Mathematics, Rice University, Houston, Texas 77005
Address at time of publication: Department of Mathematical Sciences, DePaul University, Chicago, Illinois 60614
Email: idu@rice.edu, iugarcov@depaul.edu

DOI: https://doi.org/10.1090/S0273-0979-06-01115-3
Keywords: Modular surface, geodesic flow, continued fractions, Markov partition
Received by editor(s): February 27, 2005
Received by editor(s) in revised form: January 24, 2006
Published electronically: October 2, 2006
Additional Notes: This paper is based on the AWM Emmy Noether Lecture given by the first author at the Joint Mathematics Meetings in January 2004 in Phoenix, AZ
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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