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Bulletin of the American Mathematical Society

Published by the American Mathematical Society, the Bulletin of the American Mathematical Society (BULL) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-9485 (online) ISSN 0273-0979 (print)

The 2020 MCQ for Bulletin of the American Mathematical Society is 0.47.

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Symbolic dynamics for the modular surface and beyond
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by Svetlana Katok and Ilie Ugarcovici PDF
Bull. Amer. Math. Soc. 44 (2007), 87-132 Request permission


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
  • MR Author ID: 99110
  • Email:
  • 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:,
  • 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
  • © Copyright 2006 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Bull. Amer. Math. Soc. 44 (2007), 87-132
  • MSC (2000): Primary 37D40, 37B40, 20H05
  • DOI:
  • MathSciNet review: 2265011