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Lipschitz distributions and Anosov flows

Author: Slobodan Simic
Journal: Proc. Amer. Math. Soc. 124 (1996), 1869-1877
MSC (1991): Primary 34C35, 58A30; Secondary 53C12
MathSciNet review: 1328378
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Abstract: We show that if a distribution is locally spanned by Lipschitz vector fields and is involutive a.e., then it is uniquely integrable giving rise to a Lipschitz foliation with leaves of class $ C^{1, \text {Lip}} $. As a consequence, we show that every codimension-one Anosov flow on a compact manifold of dimension $ >3 $ such that the sum of its strong distributions is Lipschitz, admits a global cross section.

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

  • [An] D.V. Anosov: Geodesic flows on closed Riemannian manifolds with negative curvature, Proceedings of the Steklov Math. Inst., no. 90 (1967), AMS Translations (1969) MR 39:3527
  • [EG] L.C. Evans and R.F. Gariepy: Measure theory and fine properties of functions, Studies in Advanced Math., CRC Press, Boca Raton, 1992 MR 93f:28001
  • [Fr] J. Franks: Anosov diffeomorphisms, Proc. Symp. in Pure Math., vol. XIV (1970), AMS MR 42:6871
  • [Gh] E. Ghys: Codimension one Anosov flows and suspensions, Lecture Notes in Math., vol. 1331 (1989), pp.59--72, Springer-Verlag
  • [Ha] P. Hartman: Ordinary differential equations, Baltimore, 1973 MR 49:9294
  • [Hi] E. Hille: Lectures on ordinary differential equations, Addison--Wesley, Reading, MA, 1969 MR 40:2939
  • [HP] M.W. Hirsch, C.C. Pugh: Stable manifolds and hyperbolic sets, Proc. Symp. in Pure Math., vol. 14 (1970), pp.133--163, AMS, Providence MR 42:6872
  • [Nh] S. Newhouse: On codimension one Anosov diffeomorphisms, American J. of Math., vol. 92 (1970), pp.761--770 MR 43:2741
  • [Pl] J.F. Plante: Anosov flows, American J. of Math., vol. 94 (1972), pp.729--754 MR 51:14099
  • [Ve] A. Verjovsky: Codimension one Anosov flows, Boletin de la Sociedad Matematica Mexicana (2) 19 (1974), no.2, pp.49--77 MR 55:4282
  • [Wa] F.W. Warner: Foundations of differentiable manifolds and Lie groups, GTM 94, Springer--Verlag, New York 1983 MR 84k:58001
  • [Wh] H. Whitney: Geometric integration theory, Princeton University Press, Princeton, 1957 MR 19:309c
  • [Zi] W.P. Ziemer: Weakly differentiable functions, GTM 120, Springer--Verlag, New York 1989 MR 91e:46046

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Additional Information

Slobodan Simic
Affiliation: Department of Mathematics, University of California at Berkeley, Berkeley, California 94720
Address at time of publication: Department of Mathematics (M/C 249), University of Illinois of Chicago, 851 S. Morgan Street, Chicago, Illinois 60607-7045

Keywords: Distribution, foliation, Anosov flow, cross section
Received by editor(s): December 15, 1994
Additional Notes: Part of this research was supported by the University of California Graduate Fellowship
Communicated by: Linda Keen
Article copyright: © Copyright 1996 American Mathematical Society