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

Sufficient conditions for smoothing codimension one foliations

Author: Christopher Ennis
Journal: Trans. Amer. Math. Soc. 276 (1983), 311-322
MSC: Primary 57R30; Secondary 57R10, 58F18
DOI: https://doi.org/10.1090/S0002-9947-1983-0684511-4
MathSciNet review: 684511
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Abstract: Let be a compact manifold. Let be a nonsingular vector field on , having unique integral curves through . For continuous, call whenever defined. Similarly, call .

For , a foliation of is said to be smoothable if there exist a foliation , which approximates , and a homeomorphism such that takes leaves of onto leaves of .

Definition. A transversely oriented Lyapunov foliation is a pair consisting of a codimension one foliation of and a nonsingular, uniquely integrable vector field on , such that there is a covering of by neighborhoods , , on which is described as level sets of continuous functions for which is continuous and strictly positive.

We prove the following theorems.

Theorem 1. Every transversely oriented Lyapunov foliation is smoothable to a transversely oriented Lyapunov foliation .

Theorem 2. If is a transversely oriented Lyapunov foliation, with and continuous for and , then is smoothable to a transversely oriented Lyapunov foliation .

The proofs of the above theorems depend on a fairly deep result in analysis due to F. Wesley Wilson, Jr. With only elementary arguments we obtain the version of Theorem 1.

Theorem 3. If is a transversely oriented Lyapunov foliation, with and is continuous, then is smoothable to a transversely oriented Lyapunov foliation .

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