Fubini foiled: pathological foliations from symbolic codings

In this talk, I'll present a counterintuitive construction of A. Katok (exposited by Milnor) which at first glance seems to contradict Fubini's theorem. In particular, one can build a full-measure set E in the unit square and a foliation of the square by smooth curves such that any leaf of the foliation meets E in exactly one point. If time permits, I'll also mention work of Ruelle-Wilkinson and Shub-Wilkinson that shows these sorts of pathological examples are common in non-uniformly hyperbolic dynamics.