Closed geodesics on a Riemann surface with application to the Markov spectrum

Authors:
A. F. Beardon, J. Lehner and M. Sheingorn

Journal:
Trans. Amer. Math. Soc. **295** (1986), 635-647

MSC:
Primary 11F99; Secondary 11J06, 30F25

MathSciNet review:
833700

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Abstract: This paper determines those Riemann surfaces on which each nonsimple closed geodesic has a parabolic intersection--that is, an intersection in the form of a loop enclosing a puncture or a deleted disk. An application is made characterizing the simple closed geodesic on in terms of the Markov spectrum.

The thrust of the situation is this: If we call loops about punctures or deleted disks *boundary curves*, then if the surface has "little" topology, each nonsimple closed geodesic must contain a boundary curve. But if there is "enough" topology, there are nonsimple closed geodesics not containing boundary curves.

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DOI:
https://doi.org/10.1090/S0002-9947-1986-0833700-7

Article copyright:
© Copyright 1986
American Mathematical Society