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Diversified homotopic behavior of closed orbits of some $\mathbb{R}$ -covered Anosov flows

  • SÉRGIO R. FENLEY (a1)

Abstract

We produce infinitely many examples of Anosov flows in closed $3$ -manifolds where the set of periodic orbits is partitioned into two infinite subsets. In one subset every closed orbit is freely homotopic to infinitely other closed orbits of the flow. In the other subset every closed orbit is freely homotopic to only one other closed orbit. The examples are obtained by Dehn surgery on geodesic flows. The manifolds are toroidal and have Seifert pieces and atoroidal pieces in their torus decompositions.

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[An]Anosov, D. V.. Geodesic flows on closed Riemannian manifolds with negative curvature. Proc. Steklov Inst. Math. 90 (1969).
[Ba1]Barbot, T.. Caractérization des flots d’Anosov pour les feuilletages faibles. Ergod. Th. & Dynam. Sys. 15 (1995), 247270.
[Ba2]Barbot, T.. Flots d’Anosov sur les variét és graph ées au sens de Waldhausen. Ann. Inst. Fourier (Grenoble) 46 (1996), 14511517.
[Barb-Fe]Barbot, T. and Fenley, S.. Pseudo-Anosov flows in toroidal manifolds. Geom. Topol. 17 (2013), 18771954.
[Ba-Fe]Barthelmé, T. and Fenley, S.. Knot theory of R-covered Anosov flows: homotopy versus isotopy of closed orbits. J. Topol. 7 (2014), 676696.
[Be-Pe]Benedetti, R. and Petronio, C.. Lectures on Hyperbolic Geometry (Universitext). Springer, Berlin, 1992.
[Bo-La]Bonatti, C. and Langevin, R.. Un exemple de flot d’Anosov transitif transverse à un tore et non conjugué à une suspension. Ergod. Th. & Dynam. Sys. 14 (1994), 633643.
[Br]Brittenham, M.. Essential laminations in Seifert fibered spaces. Topology 32 (1993), 6185.
[Ep]Epstein, D.. Periodic flows on 3-manifolds. Ann. of Math. (2) 95 (1972), 6688.
[Fe1]Fenley, S.. Anosov flows in 3-manifolds. Ann. of Math. (2) 139 (1994), 79115.
[Fe2]Fenley, S.. Quasigeodesic Anosov flows and homotopic properties of flow lines. J. Differential Geom. 41 (1995), 479514.
[Fe3]Fenley, S.. The structure of branching in Anosov flows of 3-manifolds. Comment. Math. Helv. 73 (1998), 259297.
[Fo-Ha]Foulon, P. and Hasselblatt, B.. Legendrian knots and nonalgebraic contact Anosov flows. Geom. Topol. 17 (2013), 12251252.
[Fr-Wi]Franks, J. and Williams, R.. Anomalous Anosov flows. Global Theory of Dynamical Systems (Lecture Notes in Mathematics, 819). Springer, Berlin, 1980.
[Fr]Fried, D.. Transitive Anosov flows and pseudo-Anosov maps. Topology 22 (1983), 299303.
[Ga]Gabai, D.. Convergence groups are Fuchsian groups. Ann. of Math. (2) 136 (1992), 447510.
[Ga-Oe]Gabai, D. and Oertel, U.. Essential laminations and 3-manifolds. Ann. of Math. (2) 130 (1989), 4173.
[Ha-Th]Handel, M. and Thurston, W.. Anosov flows on new 3-manifolds. Invent. Math. 59 (1980), 95103.
[He]Hempel, J.. 3-Manifolds (Annals of Mathematical Studies, 86). Princeton University Press, Princeton, NJ, 1976.
[Ja]Jaco, W.. Lectures on Three-Manifold Topology (Conference Board of Mathematical Society, 43). American Mathematical Society, Providence, RI, 1980.
[Ja-Sh]Jaco, W. and Shalen, P.. Seifert Fibered Spaces in 3-manifolds (Memoirs of the American Mathematical Society, 220). American Mathematical Society, Providence, RI, 1979.
[Li]Lickorish, W. B. R.. A representation of orientable combinatorial 3-manifolds. Ann. of Math. (2) 76 (1962), 531538.
[Ro]Rosenberg, H.. Foliations by planes. Topology 7 (1968), 131138.
[Th1]Thurston, W.. The Geometry and Topology of 3-Manifolds. Princeton University Lecture Notes, 1982.
[Th2]Thurston, W.. Three dimensional manifolds, Kleinian groups, and hyperbolic geometry. Bull. Amer. Math. Soc. 6 (1982), 357381.

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Diversified homotopic behavior of closed orbits of some $\mathbb{R}$ -covered Anosov flows

  • SÉRGIO R. FENLEY (a1)

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