Suppose we have a distribution, a subbundle of the tangent bundle of a manifold, and we want to know if it is integrable, that is, if it is tangent to a foliation on the manifold. If the distribution is smooth and invariant under a dynamical system we can use properties of the dynamics to prove that the distribution is integrable. In fact, the distribution can only fail to be integrable if specific resonance conditions exist for growth rates associated to the dynamics.
In this talk, I'll briefly introduce Oseledets theorem and show how it can be used to prove this result. I'll also show how this applies to partially hyperbolic dynamical systems and results about ergodicity, and how different the behavior is when the distribution is not smooth.
The talk should be accessible to anyone who knows what a Riemannian manifold is.