Chaotic foliations with Ehresmann connection

Abstract

We consider smooth codimension q foliations on n-dimensional manifolds where 0<q<n. We use Ehresmann connections as a technical tool to introduce the notion of sensitivity to initial conditions for foliations. We extend Devaney's definition of chaos for cascades to foliations with Ehresmann connection. Our main result states that sensitivity to initial conditions of a foliation with Ehresmann connection follows from topological transitivity and density of minimal sets of the foliation. Compactness both minimal sets and the ambient manifold is not assumed. The results are applied to complete Cartan foliations.