Every 3-manifold admits a structurally stable nonsingular flow with three basic sets

Yu, Bin

doi:10.1090/proc/13122

Every $3$-manifold admits a structurally stable nonsingular flow with three basic sets
HTML articles powered by AMS MathViewer

by Bin Yu PDF

Proc. Amer. Math. Soc. 144 (2016), 4949-4957 Request permission

Abstract:

This paper is devoted to proving that every closed orientable $3$-manifold admits a simple Smale flow $X_t$. Here a simple Smale flow is a structurally stable nonsingular flow whose chain recurrent set is composed of a periodic orbit attractor, a periodic orbit repeller and a transitive saddle invariant set, i.e., a saddle basic set.

References

F. Béguin, C. Bonatti, B. Yu, Building Anosov flows on three-manifolds. arXiv:1408.3951.
F. Béguin, C. Bonatti, B. Yu, Simple Smale flows on $3$-manifolds, in preparation.
Rufus Bowen, One-dimensional hyperbolic sets for flows, J. Differential Equations 12 (1972), 173–179. MR 336762, DOI 10.1016/0022-0396(72)90012-5
Marco Brunella, Separating the basic sets of a nontransitive Anosov flow, Bull. London Math. Soc. 25 (1993), no. 5, 487–490. MR 1233413, DOI 10.1112/blms/25.5.487
Joan S. Birman and R. F. Williams, Knotted periodic orbits in dynamical system. II. Knot holders for fibered knots, Low-dimensional topology (San Francisco, Calif., 1981) Contemp. Math., vol. 20, Amer. Math. Soc., Providence, RI, 1983, pp. 1–60. MR 718132, DOI 10.1090/conm/020/718132
John B. Etnyre, Lectures on open book decompositions and contact structures, Floer homology, gauge theory, and low-dimensional topology, Clay Math. Proc., vol. 5, Amer. Math. Soc., Providence, RI, 2006, pp. 103–141. MR 2249250
Sérgio R. Fenley, Anosov flows in $3$-manifolds, Ann. of Math. (2) 139 (1994), no. 1, 79–115. MR 1259365, DOI 10.2307/2946628
John M. Franks, Knots, links and symbolic dynamics, Ann. of Math. (2) 113 (1981), no. 3, 529–552. MR 621015, DOI 10.2307/2006996
John Franks, Nonsingular Smale flows on $S^3$, Topology 24 (1985), no. 3, 265–282. MR 815480, DOI 10.1016/0040-9383(85)90002-3
Elizabeth L. Haynes and Michael C. Sullivan, Simple Smale flows with a four band template, Topology Appl. 177 (2014), 23–33. MR 3258180, DOI 10.1016/j.topol.2014.08.003
John W. Morgan, Nonsingular Morse-Smale flows on $3$-dimensional manifolds, Topology 18 (1979), no. 1, 41–53. MR 528235, DOI 10.1016/0040-9383(79)90013-2
Dale Rolfsen, Knots and links, Mathematics Lecture Series, No. 7, Publish or Perish, Inc., Berkeley, Calif., 1976. MR 0515288
Michael C. Sullivan, Visually building Smale flows in $S^3$, Topology Appl. 106 (2000), no. 1, 1–19. MR 1769328, DOI 10.1016/S0166-8641(99)00069-3
Bin Yu, Lorenz like Smale flows on three-manifolds, Topology Appl. 156 (2009), no. 15, 2462–2469. MR 2546948, DOI 10.1016/j.topol.2009.07.008