Lipschitz distributions and Anosov flows

Author:
Slobodan Simic

Journal:
Proc. Amer. Math. Soc. **124** (1996), 1869-1877

MSC (1991):
Primary 34C35, 58A30; Secondary 53C12

DOI:
https://doi.org/10.1090/S0002-9939-96-03423-5

MathSciNet review:
1328378

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Abstract | References | Similar Articles | Additional Information

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 . As a consequence, we show that every codimension-one Anosov flow on a compact manifold of dimension such that the sum of its strong distributions is Lipschitz, admits a global cross section.

**[An]**D. V. Anosov,*Geodesic flows on closed Riemann manifolds with negative curvature.*, Proceedings of the Steklov Institute of Mathematics, No. 90 (1967). Translated from the Russian by S. Feder, American Mathematical Society, Providence, R.I., 1969. MR**0242194****[EG]**Lawrence C. Evans and Ronald F. Gariepy,*Measure theory and fine properties of functions*, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. MR**1158660****[Fr]**John Franks,*Anosov diffeomorphisms*, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, pp. 61–93. MR**0271990****[Gh]**E. Ghys: Codimension one Anosov flows and suspensions,*Lecture Notes in Math.*, vol.**1331**(1989), pp.59--72, Springer-Verlag**[Ha]**Philip Hartman,*Ordinary differential equations*, S. M. Hartman, Baltimore, Md., 1973. Corrected reprint. MR**0344555****[Hi]**Einar Hille,*Lectures on ordinary differential equations*, Addison-Wesley Publ. Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR**0249698****[HP]**Morris W. Hirsch and Charles C. Pugh,*Stable manifolds and hyperbolic sets*, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, pp. 133–163. MR**0271991****[Nh]**S. E. Newhouse,*On codimension one Anosov diffeomorphisms*, Amer. J. Math.**92**(1970), 761–770. MR**0277004**, https://doi.org/10.2307/2373372**[Pl]**Joseph F. Plante,*Anosov flows*, Amer. J. Math.**94**(1972), 729–754. MR**0377930**, https://doi.org/10.2307/2373755**[Ve]**Alberto Verjovsky,*Codimension one Anosov flows*, Bol. Soc. Mat. Mexicana (2)**19**(1974), no. 2, 49–77. MR**0431281****[Wa]**Frank W. Warner,*Foundations of differentiable manifolds and Lie groups*, Graduate Texts in Mathematics, vol. 94, Springer-Verlag, New York-Berlin, 1983. Corrected reprint of the 1971 edition. MR**722297****[Wh]**H. Whitney:*Geometric integration theory*, Princeton University Press, Princeton, 1957 MR**19:309c****[Zi]**William P. Ziemer,*Weakly differentiable functions*, Graduate Texts in Mathematics, vol. 120, Springer-Verlag, New York, 1989. Sobolev spaces and functions of bounded variation. MR**1014685**

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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

Email:
simic@math.uic.edu

DOI:
https://doi.org/10.1090/S0002-9939-96-03423-5

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