C^1,0 foliation theory

Transverse 1-dimensional foliations play an important role in the study of codimensionone foliations. In Geom. Topol. Monogr. 19 (2015) 21-72, the authors introduced the notion of flow box decomposition of a 3-manifold M. This is a combinatorial decomposition of M that reflects both the structure of a given codimension-one foliation and that of a given transverse dimension-one foliation, and that is amenable to inductive strategies. In this paper, flow box decompositions are used to extend some classical foliation results to foliations that are not C². Enhancements of well-known results of Calegari on smoothing leaves, Dippolito on Denjoy blowup of leaves, and Tischler on approximations by fibrations are obtained. The methods developed are not intrinsically 3-dimensional techniques, and should generalize to prove corresponding results for codimension-one foliations in n-dimensional manifolds.