A Simple Example of Pathological Foliations in Skew-Product Diffeomorphisms

TL;DR

This paper constructs simple, volume-preserving skew-product diffeomorphisms with pathological center foliations, where typical leaves intersect a full measure set in at most one point, highlighting a novel elementary mechanism.

Contribution

It introduces a straightforward and elementary method to produce skew-product diffeomorphisms with pathological center foliations, expanding understanding of such phenomena.

Findings

Constructed explicit examples of volume-preserving skew-product diffeomorphisms

Demonstrated the existence of full measure sets intersecting leaves minimally

Provided a simple mechanism for creating pathological foliations

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.