Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Local and global properties of limit sets of foliations of quasigeodesic Anosov flows

Author: Sérgio R. Fenley
Journal: Trans. Amer. Math. Soc. 350 (1998), 3923-3941
MSC (1991): Primary 57R30, 58F25, 58F15; Secondary 58F22, 53C12
MathSciNet review: 1432199
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A nonsingular flow is quasigeodesic when all flow lines are efficient in measuring distances in relative homotopy classes. We analyze the quasigeodesic property for Anosov flows in $3$-manifolds which have negatively curved fundamental group. We show that this property implies that limit sets of stable and unstable leaves (in the universal cover) vary continuously in the sphere at infinity. It also follows that the union of the limit sets of all stable (or unstable) leaves is not the whole sphere at infinity. Finally, under the quasigeodesic hypothesis we completely determine when limit sets of leaves in the universal cover can intersect. This is done by determining exactly when flow lines in the universal cover share an ideal point.

References [Enhancements On Off] (What's this?)

  • [An] D. V. Anosov, Geodesic flows on closed Riemannian manifolds with negative curvature, Proc. Steklov Inst. Math. 90 (1969). MR 39:3527
  • [Ba1] T. Barbot, Caractérization des flots d'Anosov en dimension 3 par leurs feuilletages faibles, Erg. Th. Dyn. Sys. 15 (1995) 247-270. MR 96d:58100
  • [Ba2] T. Barbot, Flots d'Anosov sur les variétés graphées au sens de Waldhausen, Ann. Inst. Fourier (Grenoble) 46 (1996), 1451-1517. MR 97j:57031
  • [Be-Me] M. Bestvina and J. Mess, The boundary of negatively curved groups, Jour. Amer. Math. Soc. 4 (1991) 469-481. MR 93j:20076
  • [Bo1] R. Bowen, Periodic orbits for hyperbolic flows, Amer. Jour. of Math. 94 (1972) 1-30. MR 45:7749
  • [Bo2] R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphims, Lecture Notes in Mathematics 470 Springer Verlag, 1975. MR 56:1374
  • [Ca-Th] J. Cannon and W. Thurston, Group invariant Peano curves, to appear.
  • [Ch] J. Christy, Intransitive Anosov flows on $3$-manifolds, to appear in C.B.M.S. lecture series.
  • [Fe1] S. Fenley, Asymptotic properties of depth one foliations in hyperbolic 3-manifolds, Jour. Diff. Geom. 36 (1992) 269-313. MR 93k:57030
  • [Fe2] S. Fenley, Quasi-isometric foliations, Topology 31 (1992) 667-676. MR 94a:57044
  • [Fe3] S. Fenley, Anosov flows in $3$-manifolds, Ann. of Math. 139 (1994) 79-115. MR 94m:58162
  • [Fe4] S. Fenley, Quasigeodesic Anosov flows and homotopic properties of flow lines, Jour. Diff. Geom. 41 (1995) 479-514. MR 96f:58118
  • [Fe5] S. Fenley, Continuous extension of Anosov foliations in $3$-manifolds with negatively curved fundamental group, to appear in Pacific J. Math.
  • [Fe6] S. Fenley, Homotopic indivisibility of closed orbits of Anosov flows in dimension $3$, Math. Zeit. 225 (1997), 289-294. CMP 97:16
  • [Fe7] S. Fenley, One sided branching in Anosov foliations, Comm. Math. Helv. 70 (1995) 248-266. MR 96c:57052
  • [Fe8] S. Fenley, The structure of branching in Anosov flows of $3$-manifolds, Comment. Math. Helv. (1998).
  • [Fe-Mo] S. Fenley and L. Mosher, Quasigeodesic flows in hyperbolic $3$-manifolds, preprint.
  • [Gh] E. Ghys, Flots d'Anosov sur les $3$-variétés fibrés en cercles, Ergod. Th. Dyn. Sys. 4 (1984) 67-80. MR 86b:58098
  • [Gh-Ha] E. Ghys and P. de la Harpe, eds., Sur les groupes hyperboliques d'aprés Mikhael Gromov, Progress in Math., 83 Birkháuser, 1991. MR 92f:53050
  • [Go] S. Goodman, Dehn surgery and Anosov flows, in Proceedings of the geometric dynamics conference, Lecture notes in mathematics 1007, Springer, 1983, 300-307. MR 84m:58003
  • [Gr] M. Gromov, Hyperbolic groups, in Essays on group theory, 75-263, Springer, 1987. MR 89e:20070
  • [Mor] J. Morgan, On Thurston's uniformization theorem for 3 dimensional manifolds, in The Smith conjecture, ed. by J. Morgan and H. Bass, Academic Press, 1984, pp. 37-125. MR 86i:57002
  • [Mo1] L. Mosher, Dynamical systems and the homology norm of a $3$-manifold I. Efficient interesection of surfaces and flows, Duke Math. Jour. 65 449-500 (1992). MR 93g:57018a
  • [Mo2] L. Mosher, Dynamical systems and the homology norm of a $3$-manifold II, Invent. Math. 107 243-281 (1992). MR 93g:57018b
  • [Mo3] L. Mosher, Examples of quasigeodesic flows on hyperbolic $3$-manifolds, Topology '90 (Columbus, OH, 1990), Ohio State Univ. Math. Res. Inst. Publ., vol. 1, de Gruyter, Berlin, 1992, pp. 227-241. MR 93i:58120
  • [No] S. P. Novikov, Topology of foliations, Trans. Moscow Math. Soc. 14 (1963) 268-305. MR 34:824
  • [Pa] F. Palmeira, Open manifolds foliated by planes, Ann. of Math., 107 (1978) 109-131. MR 58:18490
  • [Pl1] J. Plante, Anosov flows, Amer. J. of Math., 94 (1972) 729-754. MR 51:14099
  • [Pl2] J. Plante, Anosov flows, transversely affine foliations and a conjecture of Verjovsky, J. London Math. Soc. (2) 23 (1981) 359-262. MR 82g:58069
  • [Pl3] J. Plante, Solvable groups acting on the line, Trans. A. M. S. 278 (1983) 401-414. MR 85b:57048
  • [Sm] S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967) 747-817. MR 37:3598
  • [Su] D. Sullivan, Cycles for the dynamical study of foliated manifolds and complex manifolds, Invent. Math. 36 (1976) 225-255. MR 55:6440
  • [Th1] W. Thurston, The geometry and topology of 3-manifolds, Princeton University Lecture Notes, 1982.
  • [Th2] W. Thurston, 3-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.) 6 (1982) 357-381. MR 83h:57019

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 57R30, 58F25, 58F15, 58F22, 53C12

Retrieve articles in all journals with MSC (1991): 57R30, 58F25, 58F15, 58F22, 53C12

Additional Information

Sérgio R. Fenley
Affiliation: Department of Mathematics, Washington University, St. Louis, Missouri 63130; Department of Mathematics, Princeton University, Princeton, New Jersey 08544-1000

Received by editor(s): December 18, 1995
Received by editor(s) in revised form: November 11, 1996
Additional Notes: Reseach supported by NSF grants DMS-9201744 and an NSF postdoctoral fellowship
Article copyright: © Copyright 1998 American Mathematical Society