Sufficient conditions for smoothing codimension one foliations

Abstract

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