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Coding of geodesics and Lorenz-like templates for some geodesic flows

Published online by Cambridge University Press:  20 September 2016

PIERRE DEHORNOY  and
TALI PINSKY
PIERRE DEHORNOY
Affiliation:
University of Grenoble Alpes, IF, F-38000 Grenoble, France email pierre.dehornoy@ujf-grenoble.fr CNRS, IF, F-38000 Grenoble, France
TALI PINSKY
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BC, Canada email tali@math.ubc.ca
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Abstract

We construct a template with two ribbons that describes the topology of all periodic orbits of the geodesic flow on the unit tangent bundle to any sphere with three cone points with hyperbolic metric. The construction relies on the existence of a particular coding with two letters for the geodesics on these orbifolds.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 3 , May 2018 , pp. 940 - 960
Copyright
© Cambridge University Press, 2016 

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References

Adler, R. and Flatto, L.. Geodesic flows, interval maps, and symbolic dynamics. Bull. Amer. Math. Soc. 25 (1991), 229–334.CrossRefGoogle Scholar
Alvarez, A., Ghys, É. and Leys, J.. Dimensions, 2008. Ch. 8, http://www.dimensions-math.org/Dim_CH7_E.htm.Google Scholar
Anosov, D. V.. Geodesic flows on closed Riemannian manifolds with negative curvature. Proc. Steklov Inst. Math. 90 (1967), 235 pp.Google Scholar
Bowen, R. and Series, C.. Markov maps associated with Fuchsian groups. Publ. Math. Inst. Hautes Études Sci. 50 (1979), 153–170.CrossRefGoogle Scholar
Birman, J. S. and Williams, R. F.. Knotted periodic orbits in dynamical systems–I: Lorenz’s equations. Topology 22 (1983), 47–82 (erratum at www.math.columbia.edu/ jb/bw-KPO-I-erratum.pdf).CrossRefGoogle Scholar
Busemann, H.. The Geometry of Geodesics (Dover Books on Mathematics, 6) . Dover Publications, New York, 1985.Google Scholar
Coven, E. M. and Nitecki, Z. H.. On the genesis of symbolic dynamics as we know it. Colloq. Math. 110 (2008), 227–242.CrossRefGoogle Scholar
Dehornoy, P.. Geodesic flow, left-handedness, and templates. Algebr. Geom. Topol. 15 (2015), 1525–1597.CrossRefGoogle Scholar
Dehornoy, P.. Which geodesic flows are left-handed? Preprint, 2015, arXiv:1501.02909.Google Scholar
de Melo, W. and van Strien, S.. One-Dimensional Dynamics. Springer, Berlin, 1993, p. 605.CrossRefGoogle Scholar
Fried, D.. Transitive Anosov flows and pseudo-Anosov maps. Topology 22 (1983), 299–303.CrossRefGoogle Scholar
Ghys, É.. Knots and dynamics. Proceedings of the International Congress of Mathematicians, I. European Mathematical Society, Zürich, 2007, pp. 247–277.Google Scholar
Ghys, É.. Right-handed vector fields and the Lorenz attractor. Jpn. J. Math. 4 (2009), 47–61.CrossRefGoogle Scholar
Hadamard, J.. Les surfaces à courbures opposées et leurs lignes géodésiques. J. Math. Pures Appl. 4 (1898), 27–74.Google Scholar
Hubbard, J. H. and Sparrow, C. T.. The classification of topologically expansive Lorenz maps. Comm. Pure Appl. Math. 43 (1990), 431–443.CrossRefGoogle Scholar
Katok, S.. Fuchsian groups, geodesic flows on surfaces of constant negative curvature and symbolic coding of geodesics. Clay Math. Proc. 10 (2008), 244–320.Google Scholar
Los, J.. Volume entropy for surface groups via Bowen–Series like maps. J. Topology, 7 (2014), 120–154.CrossRefGoogle Scholar
Morse, M. and Hedlund, G. A.. Symbolic dynamics. Amer. J. Math. 60 (1938), 815–866.CrossRefGoogle Scholar
Montesinos-Amilibia, J. M.. Classical Tesselations and Three-Manifolds (Universitext) . Springer, Berlin, 1987, p. 230.CrossRefGoogle Scholar
Pfeiffer, M.. Automata and growth functions for the triangle groups. Diploma Thesis, Rheinisch-Westfalische Technische Hochschule Aachen, 2008.Google Scholar
Pinsky, T.. Templates for the geodesic flow. Ergod. Th. & Dynam. Sys. 34 (2014), 211–235.CrossRefGoogle Scholar
Pit, V.. Codage du flot géodésique sur les surfaces hyperboliques de volume fini. PhD Thesis, Bordeaux, 2010.Google Scholar
Ratner, M.. Markov partitions for Anosov flows on n-dimensional manifolds. Israel J. Math. 15 (1973), 92–114.CrossRefGoogle Scholar
Rolfsen, D.. Knots and Links. Publish or Perish, Houston, TX, 1976, p. 439.Google Scholar
Series, C.. The infinite word problem and limit sets in Fuchsian groups. Ergod. Th. & Dynam. Sys. 1 (1981), 337–360.CrossRefGoogle Scholar
Thurston, W.. On the geometry and dynamics of diffeomorphisms of surfaces. Bull. Amer. Math. Soc. 19 (1988), 417–431.CrossRefGoogle Scholar