Bizarre topology is natural in dynamical systems
We describe an example of a $C^\infty$ diffeomorphism on a 7--manifold which has a compact invariant set such that uncountably many of its connected components are pseudocircles. (Any 7--manifold will suffice.) Furthermore, any diffeomorphism which is sufficiently close (in the $C^1$ metric) to the constructed map has a similar invariant set, and the dynamics of the map on the invariant set are chaotic.
math.DS↗