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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Sufficient conditions for smoothing codimension one foliations
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by Christopher Ennis PDF
Trans. Amer. Math. Soc. 276 (1983), 311-322 Request permission

Abstract:

Let $M$ be a compact ${C^\infty }$ manifold. Let $X$ be a ${C^0}$ nonsingular vector field on $M$, having unique integral curves $(p,t)$ through $p \in M$. For $f: M \to {\mathbf {R}}$ continuous, call $\left . Xf(p) = df(p,t)/dt\right |_{t = 0}$ whenever defined. Similarly, call ${X^k}f(p)=X(X^{k-1}f)(p)$. For $0 \leqslant r < k$, a ${C^r}$ foliation $\mathcal {F}$ of $M$ is said to be ${C^k}$ smoothable if there exist a ${C^k}$ foliation $\mathcal {G}$, which ${C^r}$ approximates $\mathcal {F}$, and a homeomorphism $h:M \to M$ such that $h$ takes leaves of $\mathcal {F}$ onto leaves of $\mathcal {G}$. Definition. A transversely oriented Lyapunov foliation is a pair $(\mathcal {F},X)$ consisting of a ${C^0}$ codimension one foliation $\mathcal {F}$ of $M$ and a ${C^0}$ nonsingular, uniquely integrable vector field $X$ on $M$, such that there is a covering of $M$ by neighborhoods $\{{W_i}\}$, $0 \leqslant i \leqslant N$, on which $\mathcal {F}$ is described as level sets of continuous functions ${f_i}:{W_i} \to {\mathbf {R}}$ for which $X{f_i}(p)$ is continuous and strictly positive. We prove the following theorems. Theorem 1. Every ${C^0}$ transversely oriented Lyapunov foliation $(\mathcal {F},X)$ is ${C^1}$ smoothable to a ${C^1}$ transversely oriented Lyapunov foliation $(\mathcal {G},X)$. Theorem 2. If $(\mathcal {F},X)$ is a ${C^0}$ transversely oriented Lyapunov foliation, with $X \in {C^{k - 1}}$ and ${X^j}{f_i}(p)$ continuous for $1 \leqslant j \leqslant k$ and $0 \leqslant i \leqslant N$, then $(\mathcal {F},X)$ is ${C^k}$ smoothable to a ${C^k}$ transversely oriented Lyapunov foliation $(\mathcal {G},X)$. 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 ${C^k}$ version of Theorem 1. Theorem 3. If $(\mathcal {F},X)$ is a ${C^{k - 1}}\;(k \geqslant 2)$ transversely oriented Lyapunov foliation, with $X \in {C^{k - 1}}$ and ${X^k}{f_i}(p)$ is continuous, then $(\mathcal {F},X)$ is ${C^k}$ smoothable to a ${C^k}$ transversely oriented Lyapunov foliation $(\mathcal {G},X)$.
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Additional Information
  • © Copyright 1983 American Mathematical Society
  • 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