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Templates for geodesic flows

Published online by Cambridge University Press:  28 November 2012

TALI PINSKY
TALI PINSKY*
Affiliation:
The Technion, Israeli Institute of Technology, Mathematics Department, Haifa 32000, Israel (email: [email protected])
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Abstract

We construct templates for geodesic flows on an infinite family of Hecke triangle groups. Our results generalize those of E. Ghys [Knots and dynamics. Proc. Int. Congress of Mathematicians. Vol. 1. International Congress of Mathematicians, Zürich, 2007], who constructed a template for the modular flow in the complement of the trefoil knot in $S^3$. A significant difficulty that arises in any attempt to go beyond the modular flow is the fact that for other Hecke triangles the geodesic flow cannot be viewed as a flow in $S^3$, and one is led to consider embeddings into lens spaces. Our final result is an explicit description of a single ‘Hecke template’ which contains all other templates we construct, allowing a topological study of the periodic orbits of different Hecke triangle groups all at once.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 34 , Issue 1 , February 2014 , pp. 211 - 235
Copyright
Copyright © 2012 Cambridge University Press 

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References

[1]Berry, M. V.. Quantum theory near the classical limit. Highlights Phys. (SERC) (1990), 50–51.Google Scholar
[2]Birman, J.. Book review: Braid groups, ordering braids. Bull. Amer. Math. Soc. 48(1) (2011), 137–146.CrossRefGoogle Scholar
[3]Birman, J. and Williams, R. F.. Knotted periodic orbits in dynamical systems–I: Lorenz’s equations. Topology 22(1) (1983), 47–82.CrossRefGoogle Scholar
[4]Birman, J. and Williams, R. F.. Knotted periodic orbits in dynamical systems-II: Knot holders for fibered knots. Cont. Math. 20 (1983), l–60.Google Scholar
[5]Cvitanovic, P.. Dynamical averaging in terms of periodic orbits. Physica D 83 (1995), 109–123.Google Scholar
[6]Dehornoy, P.. Les noeuds de Lorenz. L’enseignement Math. (2) 57 (2011), 211–280.CrossRefGoogle Scholar
[7]Dehornoy, P.. Geodesic flow, left-handedness, and templates. Preprint, 2011, arXiv:1112.6296.Google Scholar
[8]Ghrist, R. W., Holmes, P. J. and Sullivan, M. C.. Knots and Links in Three-Dimensional Flows. Princeton University Press, Princeton, NJ, 1997.Google Scholar
[9]Ghrist, R. W.. Branched two-manifolds supporting all links. Topology 36 (1997), 423–448.Google Scholar
[10]Ghys, E.. Knots and dynamics. Proc. Int. Congress of Mathematicians. Vol. 1. International Congress of Mathematicians, Zürich, 2007.Google Scholar
[11]Mackenzie, D.. A new twist in knot theory. What’s Happening Math. Sci. 7 (2009), 2–17.Google Scholar
[12]Montesinos, J. M.. Classical Tessellations and Three-Manifolds. Springer, 1987.Google Scholar
[13]Pinsky, T.. Primeness of Hecke knots, in preparation.Google Scholar
[14]Rolfsen, D.. Knots and Links. American Mathematical Society, Providence, RI, 2003.Google Scholar
[15]Seifert, H.. Topology of 3-dimensional fibered spaces. Acta. Math. 60 (1933), 147–288.CrossRefGoogle Scholar
[16]Steenrod, N.. The Topology of Fiber Bundles. Princeton University Press, Princeton, NJ, 1951.Google Scholar
[17]Sullivan, M.. Prime decomposition of knots in Lorenz-like templates. J. Knot Theory Ramifications 2(4) (1993), 453–462.Google Scholar
[18]Sullivan, M.. Factoring families of positive knots on Lorenz-like templates. J. Knot Theory Ramifications 17(10) (2008).Google Scholar
[19]Tsanov, V. V.. Triangle groups, automorphic forms, and torus knots. Preprint, arXiv:1011.0461v2.Google Scholar
[20]Watkins, M.. A short survey of lens spaces. Undergraduate dissertation, University of Kent at Canterbury, 1990.Google Scholar