Computing selfintersections of closed geodesics on finitesheeted covers of the modular surface
Authors:
J. Lehner and M. Sheingorn
Journal:
Math. Comp. 44 (1985), 233240
MSC:
Primary 11F06; Secondary 11J06, 20H10, 30F35
MathSciNet review:
771045
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Abstract: An algorithm is given for deciding whether a closed geodesic on a finitesheeted cover of the modular surface has selfintersections; if it does, the algorithm gives them in the order they occur in traversing the geodesic. The following general result on geodesics is proved: any closed geodesic on a Riemann surface R can be lifted to a simple closed geodesic on some finitesheeted cover of R. In the last two sections the connection with the stabilizer (under the modular group) of a Markov quadratic irrationality is discussed.
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 A. F. Beardon, J. Lehner & M. Sheingorn, "Closed simple geodesics on Riemann surfaces and the Markov spectrum." (To be published.)
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 J. Birman & C. Series, "An algorithm for simple curves on surfaces," J. London Math. Soc. (2), v. 29, 1984, pp. 331342. MR 744104 (85m:57002)
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 J. Birman & C. Series, "Simple curves have Hausdorff dimension one." (Preprint.)
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 J. W. S. Cassels, An Introduction to Diophantine Approximation, Cambridge Univ. Press, Cambridge, 1957. MR 0087708 (19:396h)
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 J. F. Koksma, Diophantische Approximationen, SpringerVerlag, Berlin, 1936; reprinted, Chelsea, New York. MR 0344200 (49:8940)
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 D. Zagier, "On the number of Markoff numbers below a given bound," Math. Comp., v. 39, 1982, pp. 709723. MR 669663 (83k:10062)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198507710457
PII:
S 00255718(1985)07710457
Article copyright:
© Copyright 1985
American Mathematical Society
