A Simple Example of Pathological Foliations in Skew-Product Diffeomorphisms

Abstract

Inspired by examples of Katok and Milnor \cite{Milnor1997}, we construct simple examples of skew-product volume preserving diffeomorphism where the center foliation is pathological in the sense that, there is a full measure set whose intersection with any center leaf contains at most one point. Comparing with other examples in literature, our mechanism for producing such pathological foliation is simple and elementary.