A pathological area preserving $C^{\infty }$ diffeomorphism of the plane

Abstract:

The pseudocircle $P$ is an hereditarily indecomposable planar continuum. In particular, it is connected but nowhere locally connected. We construct a ${C^\infty }$ area preserving diffeomorphism of the plane with $P$ as a minimal set. The diffeomorphism $f$ is constructed as an explicit limit of diffeomorphisms conjugate to rotations about the origin. There is a well-defined irrational rotation number for $f|P$ even though $f|P$ is not even semi-conjugate to a rotation of ${S^1}$. If we remove the requirement that our diffeomorphisms be area preserving, we may alter the example so that $P$ is an attracting minimal set.