×

Plane affine geometry and Anosov flows. (English) Zbl 1098.37513

Summary: We study Anosov flows on closed 3-manifolds. We define the notion of Anosov flows with the topological contact property (abreviation TCP Anosov flows): typical examples of TCP Anosov flows are contact Anosov flows, i.e. flows preserving a contact 1-form. We show that TCP Anosov flows are \(\mathbb R\)-covered. The main tool is the study of the leaf spaces of lifted strong stable foliations: we exhibit on these leaf spaces a structure of (generalized) affine plane, in the sense of incidence geometry.

MSC:

37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
57R30 Foliations in differential topology; geometric theory
37E99 Low-dimensional dynamical systems
PDFBibTeX XMLCite
Full Text: DOI Numdam EuDML

References:

[1] Anosov D.V. , Geodesic flows on closed Riemannian manifolds of negative curvature , Trudy Mat. Inst. Steklov. 90 ( 1967 ). MR 224110 | Zbl 0176.19101 · Zbl 0176.19101
[2] Barbot T. , Caractérisation des flots d’Anosov en dimension 3 par leurs feuilletages faibles , Ergodic Theory Dynam. Systems 15 ( 1995 ) 247 - 270 . MR 1332403 | Zbl 0826.58025 · Zbl 0826.58025
[3] Barbot T. , Flots d’Anosov sur les variétés graphées au sens de Waldhausen , Ann. Inst. Fourier (Grenoble) 46 ( 1996 ) 1451 - 1517 . Numdam | MR 1427133 | Zbl 0861.58028 · Zbl 0861.58028
[4] Barbot T. , Generalizations of the Bonatti-Langevin example of Anosov flow and their classification up to topological equivalence , Comm. Anal. Geom. 6 ( 1998 ) 749 - 798 . MR 1652255 | Zbl 0916.58033 · Zbl 0916.58033
[5] Bonatti C. , Langevin R. , Un exemple de flot d’Anosov transitif transverse à un tore et non conjugué à une suspension , Ergodic Theory Dynam. Systems 14 ( 1994 ) 633 - 643 . MR 1304136 | Zbl 0826.58026 · Zbl 0826.58026
[6] Bowen R. , Marcus B. , Unique ergodicity for horocycle foliations , Israel J. Math. 26 ( 1 ) ( 1977 ) 43 - 67 . MR 451307 | Zbl 0346.58009 · Zbl 0346.58009
[7] Buekenhout F. , Handbook of Incidence Geometry , North-Holland , Amsterdam , 1995 , Edited by F. Buekenhout, 1420 pp. MR 1360715 | Zbl 0821.00012 · Zbl 0821.00012
[8] Fenley S.R. , Anosov flows in 3-manifolds , Ann. of Math. (2) 139 ( 1 ) ( 1994 ) 79 - 115 . MR 1259365 | Zbl 0796.58039 · Zbl 0796.58039
[9] Fenley S.R. , The structure of branching in Anosov flows of 3-manifolds , Comment. Math. Helv. 73 ( 2 ) ( 1998 ) 259 - 297 . MR 1611703 | Zbl 0999.37008 · Zbl 0999.37008
[10] Foulon P ., private communication.
[11] Franks J. , Anosov diffeomorphisms , in: Global Analysis (Berkeley, Calif., 1968) , Proc. Sympos. Pure Math. , XIV , American Mathematical Society , Providence, RI , 1970 , pp. 61 - 93 . MR 271990 | Zbl 0207.54304 · Zbl 0207.54304
[12] Franks J. , Williams B. , Anomalous Anosov flows , in: Lectures Notes in Math. , 819 , 1980 , pp. 158 - 174 . MR 591182 | Zbl 0463.58021 · Zbl 0463.58021
[13] Fried D. , Transitive Anosov flows and pseudo-Anosov maps , Topology 22 ( 1983 ) 299 - 303 . MR 710103 | Zbl 0516.58035 · Zbl 0516.58035
[14] Ghys E. , Flots d’Anosov sur les 3-variétés fibrées en cercles , Ergodic Theory Dynam. Systems 4 ( 1 ) ( 1984 ) 67 - 80 . MR 758894 | Zbl 0527.58030 · Zbl 0527.58030
[15] Ghys E. , Flots d’Anosov dont les feuilletages stables sont différentiables , Ann. Sci. École Norm. Sup. (4) 20 ( 2 ) ( 1987 ) 251 - 270 . Numdam | MR 911758 | Zbl 0663.58025 · Zbl 0663.58025
[16] Ghys E. , Déformations de flots d’Anosov et de groupes fuchsiens , Ann. Inst. Fourier (Grenoble) 42 ( 1-2 ) ( 1992 ) 209 - 247 . Numdam | MR 1162561 | Zbl 0759.58036 · Zbl 0759.58036
[17] Ghys E. , Rigidité différentiable des groupes fuchsiens , Inst. Hautes Études Sci. Publ. Math. 78 ( 1993 ) 163 - 185 . Numdam | MR 1259430 | Zbl 0812.58066 · Zbl 0812.58066
[18] Goodman S. , Dehn surgery on Anosov flows , in: Lectures Notes in Math. , 1007 , 1983 , pp. 300 - 307 . MR 1691596 | Zbl 0532.58021 · Zbl 0532.58021
[19] Handel M. , Thurston W. , Anosov flows on new three manifolds , Invent. Math. 59 ( 1980 ) 95 - 103 . MR 577356 | Zbl 0435.58019 · Zbl 0435.58019
[20] Hasselblatt B. , Katok A. , Introduction to the Modern Theory of Dynamical Systems (With a supplementary chapter by A. Katok and L. Mendoza) , Encyclopedia of Mathematics and its Applications , 54 , Cambridge University Press , Cambridge , 1995 . MR 1326374 | Zbl 0878.58020 · Zbl 0878.58020
[21] Hasselblatt B. , Wilkinson A. , Prevalence of non-Lipschitz Anosov foliations , Ergodic Theory Dynam. Systems 19 ( 1998 ) 643 - 656 . MR 1695913 | Zbl 1069.37031 · Zbl 1069.37031
[22] Hirsch M.W. , Pugh C. , Stable manifolds and hyperbolic sets , in: Global Analysis (Berkeley, Calif., 1968) , Proc. Sympos. Pure Math. , XIV , American Mathematical Society , Providence, RI , 1970 , pp. 133 - 163 . MR 271991 | Zbl 0215.53001 · Zbl 0215.53001
[23] Hurder S. , Katok A. , Differentiability, rigidity and Godbillon-Vey classes for Anosov flows , Inst. Hautes Études Sci. Publ. Math. 72 ( 1990 ) 5 - 61 . Numdam | MR 1087392 | Zbl 0725.58034 · Zbl 0725.58034
[24] Margulis G.A. , Certain measures that are connected with U -flows on compact manifolds , Functional Anal. Appl. 4 ( 1970 ) 55 - 67 . MR 272984 | Zbl 0245.58003 · Zbl 0245.58003
[25] Newhouse S.E. , On codimension one Anosov diffeomorphisms , Amer. J. Math. 92 ( 1970 ) 761 - 770 . MR 277004 | Zbl 0204.56901 · Zbl 0204.56901
[26] Palmeira C.F.B. , Open manifolds foliated by planes , Ann. Math. 107 ( 1978 ) 109 - 131 . MR 501018 | Zbl 0382.57010 · Zbl 0382.57010
[27] Plante J.F. , Anosov flows , Amer. J. Math. 94 ( 1972 ) 729 - 754 . MR 377930 | Zbl 0257.58007 · Zbl 0257.58007
[28] Plante J.F. , Anosov flows, transversely affine foliations, and a conjecture of Verjovsky , J. London Math. Soc. (2) 23 ( 2 ) ( 1981 ) 359 - 362 . MR 609116 | Zbl 0465.58020 · Zbl 0465.58020
[29] Plante J.F. , Thurston W. , Anosov flows and the fundamental group , Topology 11 ( 1972 ) 147 - 150 . MR 295389 | Zbl 0246.58014 · Zbl 0246.58014
[30] Salzmann H. , Betten D. , Grundhöfer T. , Hähl H. , Löwen R. , Stroppel M. , Compact Projective Planes , De Gruyter Expositions in Mathematics , 21 , Walter de Gruyter , Berlin , 1995 . MR 1384300 | Zbl 0851.51003 · Zbl 0851.51003
[31] Simić S. , Codimension one Anosov flows and a conjecture of Verjovsky , Ergodic Theory Dynam. Systems 17 ( 1997 ) 1221 - 1231 . MR 1477039 | Zbl 0903.58026 · Zbl 0903.58026
[32] Solodov V.V. , The universal cover of Anosov flows , preprint , 1992 .
[33] Thurston W. , Three-manifolds, foliations and circles, I , preprint , 1997 , math.gt/9712268. arXiv | MR 380828
[34] Verjovsky A. , Codimension one Anosov flows , Bol. Soc. Mexicana (2) 19 ( 2 ) ( 1974 ) 49 - 77 . MR 431281 | Zbl 0323.58014 · Zbl 0323.58014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.