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Global cross sections for Anosov flows

Published online by Cambridge University Press:  15 June 2015

SLOBODAN N. SIMIĆ
SLOBODAN N. SIMIĆ*
Affiliation:
Department of Mathematics and Statistics, San José State University, San José, CA 95192-0103, USA email [email protected]
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Abstract

We provide a new criterion for the existence of a global cross section to a volume-preserving Anosov flow. The criterion is expressed in terms of expansion and contraction rates of the flow and generalizes known results of this type.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 36 , Issue 8 , December 2016 , pp. 2661 - 2674
Copyright
© Cambridge University Press, 2015 

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References

Anosov, D. V.. Geodesic flows on closed Riemannian manifolds of negative curvature. Proc. Steklov Inst. Math. 90 (1967), AMS Translations, 1969.Google Scholar
Anosov, D. V. and Sinai, Y. G.. Some smooth ergodic systems. Russian Math. Surveys 22 (1967), 103–167.Google Scholar
Armendariz, P.. Codimension one Anosov flows on manifolds with solvable fundamental group. PhD Thesis, Universidad Autónoma Metropolitana, Unidad Iztapalapa, Mexico, D. F., 1982.Google Scholar
Asaoka, M.. On invariant volumes of codimension-one Anosov flows and the Verjovsky conjecture. Invent. Math. 174(2) (2008), 435–462.Google Scholar
Bonatti, C. and Guelman, N.. Transitive Anosov flows and Axiom-A diffeomorphisms. Ergod. Th. & Dynam. Sys. 29 (2009), 817–848.Google Scholar
Evans, L. C.. Partial Differential Equations (Graduate Studies in Mathematics, 19) . American Mathematical Society, Providence, RI, 1998.Google Scholar
Franks, J.. Anosov diffeomorphisms. Proc. Sympos. Pure Math. (1970), 61–93.Google Scholar
Fried, D.. The geometry of cross sections to flows. Topology 21 (1982), 353–371.Google Scholar
Ghys, E.. Codimension one Anosov Flows and Suspensions. Dynamical Systems (Lecture Notes in Mathematics, 1331) . Eds. Bamón, R., Labarca, R. and Palis, J.. Springer, Berlin, 1989, pp. 59–72.Google Scholar
Hasselblatt, B.. Regularity of the Anosov splitting and of horospheric foliations. Ergod. Th. & Dynam. Sys. 14 (1994), 645–666.Google Scholar
Hasselblatt, B.. Regularity of the Anosov splitting II. Ergod. Th. & Dynam. Sys. 17 (1997), 169–172.Google Scholar
Hasselblatt, B. and Wilkinson, A.. Prevalence of non-Lipschitz Anosov foliations. Ergod. Th. & Dynam. Sys. 19(3) (1999), 643–656.Google Scholar
Hirsch, M. W., Pugh, C. C. and Shub, M.. Invariant Manifolds (Lecture Notes in Mathematics, 583) . Springer, Berlin, 1977.Google Scholar
Katok, A. and Hasselblatt, B.. Introduction to the Modern Theory of Dynamical Systems (Encyclopedia of Mathematics and its Applications, 54) . Cambridge University Press, Cambridge, 1995.Google Scholar
Newhouse, S.. On codimension one Anosov diffeomorphisms. Amer. J. Math. 92 (1970), 761–770.CrossRefGoogle Scholar
Plante, J.. Anosov flows. Amer. J. Math. 94 (1972), 729–754.Google Scholar
Plante, J.. Anosov flows, transversely affine foliations, and a conjecture of Verjovsky. J. Lond. Math. Soc. (2) 2 (1981), 359–362.Google Scholar
Plante, J.. Solvable groups acting on the line. Trans. Amer. Math. Soc. 278(1) (1983), 401–414.Google Scholar
Pugh, C. C., Shub, M. and Wilkinson, A.. Hölder foliations. Duke Math. J. 86(3) (1997), 517–546.Google Scholar
Schwartzman, S.. Asymptotic cycles. Ann. of Math. (2) 66 (1957), 270–284.CrossRefGoogle Scholar
Sharp, R.. Closed orbits in homology classes for Anosov flows. Ergod. Th. & Dynam. Sys. 13 (1993), 387–408.Google Scholar
Simić, S. N.. Lipschitz distributions and Anosov flows. Proc. Amer. Math. Soc. 124(6) (1996), 1869–1877.Google Scholar
Simić, S. N.. Codimension one Anosov flows and a conjecture of Verjovsky. Ergod. Th. & Dynam. Sys. 17 (1997), 1211–1231.Google Scholar
Spivak, M.. A Comprehensive Introduction to Differential Geometry. Vol. 1, 3rd edn. Publish or Perish, Berkeley, CA, 2005.Google Scholar
Stein, E. M.. Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton, NJ, 1970.Google Scholar
Verjovsky, A.. Codimension one Anosov flows. Bol. Soc. Mat. Mexicana (3) 19 (1974), 49–77.Google Scholar