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Genus-one Birkhoff sections for geodesic flows

Published online by Cambridge University Press:  11 August 2014

PIERRE DEHORNOY
PIERRE DEHORNOY*
Affiliation:
Mathematisches Institut, Universität Bern, Sidlerstrasse 5, 3012 Bern, Switzerland email pierre.dehornoy@gmail.com Present address: Institut Fourier, Université Joseph Fourier-Grenoble 1, 100 rue des mathématiques, BP74, 38402 Saint-Martin-d’H\`eres cedex, France
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Abstract

We prove that the geodesic flow on the unit tangent bundle to every hyperbolic 2-orbifold that is a sphere with three or four singular points admits explicit genus-one Birkhoff sections, and we determine the associated first return maps.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 35 , Issue 6 , September 2015 , pp. 1795 - 1813
Copyright
© Cambridge University Press, 2014 

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References

A’Campo, N.. Generic immersions of curves, knots, monodromy and gordian number. Publ. Math. Inst. Hautes Études Sci. 88 (1998), 152–169.CrossRefGoogle Scholar
Anosov, D. V.. Geodesic flows on closed Riemannian manifolds with negative curvature. Proc. Steklov Inst. Math. 90 (1967), 3–210.Google Scholar
Birkhoff, G.. Dynamical systems with two degrees of freedom. Trans. Amer. Math. Soc. 18 (1917), 199–300.CrossRefGoogle Scholar
Birman, J. and Williams, R.. Knotted periodic orbits in dynamical systems II: knot-holders for fibered knots. Low Dimensional Topology (Contemporary Mathematics, 20). Ed. Lomonaco, S. Jr. American Mathematical Society, Princeton, NJ, 1981, pp. 1–60.Google Scholar
Brunella, M.. On the discrete Godbillon-Vey invariant and Dehn surgery on geodesic flows. Ann. Fac. Sci. Toulouse 3 (1994), 335–346.CrossRefGoogle Scholar
Dehornoy, Pi.. Geodesic flow, left-handedness, and templates. Preprint, arXiv:1112.6296v2.Google Scholar
Fine, B.. Trace classes and quadratic forms in the modular group. Canad. Math. Bull. 37 (1994), 202–212.CrossRefGoogle Scholar
Fried, D.. Transitive Anosov flows and pseudo-Anosov maps. Topology 22 (1983), 299–303.CrossRefGoogle Scholar
Ghys, É.. Flots d’Anosov sur les 3-variétés fibrées en cercles. Ergod. Th. & Dynam. Sys. 4 (1984), 67–80.CrossRefGoogle Scholar
Ghys, É.. Sur l’invariance topologique de la classe de Godbillon-Vey. Ann. Inst. Fourier 37 (1987), 59–76.CrossRefGoogle Scholar
Hadamard, J.. Les surfaces à courbures opposées et leurs lignes géodésiques. J. Math. Pures Appl. 4 (1898), 27–74.Google Scholar
Hashiguchi, N.. On the Anosov diffeomorphisms corresponding to geodesic flow on negatively curved closed surfaces. J. Fac. Sci. Univ. Tokyo 37 (1990), 485–494.Google Scholar
Thurston, W.. The topology and geometry of three-manifolds. Unpublished manuscript, 1980, library.msri.org/books/gt3m/.Google Scholar