  Various issues with regard to chaos and recurrence in infinite dimensions are
discussed. The doctrine we are trying to derive is that Sobolev spaces over
bounded spatial domains do host chaos and recurrence, while Sobolev spaces over
unbounded spatial domains are lack of chaos and recurrence. Local Sobolev
spaces over unbounded spatial domains can host chaos and are natural phase
spaces e.g. for fluid problems, but are very challenging to study.
