Sufficient conditions for smoothing codimension one foliations

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 $(\mathcal {G},X)$.