On montre que, dans le fibré unitaire tangent d’une orbisphère hyperbolique à points coniques d’ordres 3, 3 et 4, le complémentaire du relevé de la plus courte géodésique périodique est homéomorphe au complémentaire du nœ ud de huit dans la sphère de dimension 3. La preuve repose in fine sur des calculs d’enlacement.
@article{TSG_2019-2021__36__1_0, author = {Pierre Dehornoy}, title = {La courbe en huit sur les sph\`eres \`a pointes et le n{\oe}ud de huit}, journal = {S\'eminaire de th\'eorie spectrale et g\'eom\'etrie}, pages = {1--30}, publisher = {Institut Fourier}, address = {Grenoble}, volume = {36}, year = {2019-2021}, doi = {10.5802/tsg.371}, language = {fr}, url = {https://proceedings.centre-mersenne.org/articles/10.5802/tsg.371/} }
TY - JOUR AU - Pierre Dehornoy TI - La courbe en huit sur les sphères à pointes et le nœud de huit JO - Séminaire de théorie spectrale et géométrie PY - 2019-2021 SP - 1 EP - 30 VL - 36 PB - Institut Fourier PP - Grenoble UR - https://proceedings.centre-mersenne.org/articles/10.5802/tsg.371/ DO - 10.5802/tsg.371 LA - fr ID - TSG_2019-2021__36__1_0 ER -
%0 Journal Article %A Pierre Dehornoy %T La courbe en huit sur les sphères à pointes et le nœud de huit %J Séminaire de théorie spectrale et géométrie %D 2019-2021 %P 1-30 %V 36 %I Institut Fourier %C Grenoble %U https://proceedings.centre-mersenne.org/articles/10.5802/tsg.371/ %R 10.5802/tsg.371 %G fr %F TSG_2019-2021__36__1_0
Pierre Dehornoy. La courbe en huit sur les sphères à pointes et le nœud de huit. Séminaire de théorie spectrale et géométrie, Tome 36 (2019-2021), pp. 1-30. doi : 10.5802/tsg.371. https://proceedings.centre-mersenne.org/articles/10.5802/tsg.371/
[1] Dmitriĭ V. Anosov Geodesic flows on closed Riemannian manifolds of negative curvature, Tr. Mat. Inst. Steklova, Volume 90 (1967), p. 209 | MR | Zbl
[2] George D. Birkhoff Dynamical systems with two degrees of freedom, Trans. Am. Math. Soc., Volume 18 (1917), pp. 199-300 | DOI | MR | Zbl
[3] Joan S. Birman; R. F. Williams Knotted periodic orbits in dynamical systems. I : Lorenz’s equations, Topology, Volume 22 (1983), pp. 47-82 | DOI | MR | Zbl
[4] Rufus Bowen; Caroline Series Markov maps associated with Fuchsian groups, Publ. Math., Inst. Hautes Étud. Sci., Volume 50 (1979), pp. 153-170 | DOI | Numdam | Zbl
[5] Marco Brunella On the discrete Godbillon-Vey invariant and Dehn surgery on geodesic flow, Ann. Fac. Sc. Toulouse Sér. 6, Volume 3 (1994), pp. 335-344 | DOI | MR | Zbl
[6] Marcos Cossarini; Pierre Dehornoy Intersection norms on surfaces and Birkhoff cross sections (2016) | arXiv
[7] Pierre Dehornoy Genus one Birkhoff cross sections for geodesic flows, Ergodic Theory Dyn. Syst., Volume 35 (2015) no. 6, pp. 1795-1813 | DOI | MR | Zbl
[8] Pierre Dehornoy Which geodesic flows are left-handed ?, Groups Geom. Dyn., Volume 11 (2017) no. 4, pp. 1347-1376 | DOI | MR | Zbl
[9] Pierre Dehornoy; Burak Ozbagci Complex vs convex Morse functions and geodesic open books (2021) (to appear in International Journal of Mathematics) | arXiv
[10] Pierre Dehornoy; Ana Rechtman Vector fields and genus in dimension 3, Int. Math. Res. Not., Volume 2022 (2022) no. 5, pp. 3262-3277 | arXiv | DOI | MR | Zbl
[11] Pierre Dehornoy; Mario Shannon Almost equivalence of algebraic Anosov flows (2019) | arXiv
[12] John B. Etnyre Lectures on open book decompositions and contact structures, Floer homology, Gauge theory, and low-dimensional topology. Proceedings of the Clay Mathematics Institute 2004 summer school (Clay Mathematics Proceedings), Volume 5, American Mathematical Society ; Clay Mathematics Institute (2006), pp. 103-141 | arXiv | MR | Zbl
[13] David Fried Fibrations over S1 with pseudo-Anosov monodromy (Astérisque), Volume 66-67, Société Matématique de France, 1979, pp. 251-266 | Zbl
[14] David Fried The geometry of cross sections to flows, Topology, Volume 21 (1982), pp. 353-371 | DOI | MR | Zbl
[15] David Fried Transitive Anosov flows and pseudo-Anosov maps, Topology, Volume 22 (1983), pp. 299-303 | DOI | MR | Zbl
[16] Étienne Ghys Sur l’invariance topologique de la classe de Godbillon-Vey, Ann. Inst. Fourier, Volume 37 (1987), pp. 59-76 | DOI | Numdam | MR | Zbl
[17] Mikhaïl Gromov Three remarks on geodesic dynamics and fundamental group, Enseign. Math., Volume 46 (2000) no. 3-4, pp. 391-402 | MR | Zbl
[18] Wolfgang Haken Theorie der Normalflächen, Acta Math., Volume 105 (1961), pp. 245-375 | DOI | Zbl
[19] Norikazu Hashiguchi On the Anosov diffeomorphisms corresponding to geodesic flow on negatively curved closed surfaces, J. Fac. Sci. Univ. Tokyo, Volume 37 (1990), pp. 485-494 | DOI | MR | Zbl
[20] Norikazu Hashiguchi; Hiroyuki Minakawa Genus one Birkhoff sections for the geodesic flows of hyperbolic 2-orbifolds, Volume 72 (2017), pp. 367-386 | MR | Zbl
[21] Greg Kuperberg Knottedness is in NP, modulo GRH, Adv. Math., Volume 256 (2014), pp. 493-506 | DOI | MR | Zbl
[22] Sergei Matveev Algorithmic topology and classification of 3-manifolds, Algorithms and Computation in Mathematics, 9, Springer, 2007 | DOI | MR | Zbl
[23] Dehornoy Pierre; Tali Pinski Coding of geodesics and Lorenz-like templates for some geodesic flows, Ergodic Theory Dyn. Syst., Volume 38 (2018) no. 3, pp. 940-960 | DOI | MR | Zbl
[24] Vincent Pit Codage du flot géodésique sur les surfaces hyperboliques de volume fini, Ph. D. Thesis, Université Bordeaux 1, Bordeaux, France (2010) (http://www.ime.unicamp.br/~pit/thesis_vincent_pit.pdf)
[25] Dale Rolfsen Knots and Links, Mathematics lecture series, 346, Publish or Perish Inc., 1976 | Zbl
[26] Saveliev Invariants for homology 3-spheres, Encyclopaedia of Mathematical Sciences, 140, Springer, 2002 | DOI | Zbl
[27] Sol Schwartzman Asymptotic cycles, Ann. Math., Volume 66 (1957), pp. 270-284 | DOI | MR
[28] Mario Shannon Hyperbolic models for transitive topological Anosov flows in dimension three (2021) | arXiv
[29] Dennis Sullivan Cycles for the dynamical study of foliated manifolds and complex manifolds, Invent. Math., Volume 36 (1976), pp. 225-255 | arXiv | DOI | MR | Zbl
[30] William Thurston The geometry and topology of three-manifolds. Vol. 1, Princeton lecture notes, 35, Princeton University Press, 1997 | Zbl
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