  Mushroom billiards have the remarkable property to show one or more clear cut
integrable islands in one or several chaotic seas, without any fractal
boundaries. The islands correspond to orbits confined to the hats of the
mushrooms, which they share with the chaotic orbits. It is thus interesting to
ask how long a chaotic orbit will remain in the hat before returning to the
stem. This question is equivalent to the inquiry about delay times for
scattering from the hat of the mushroom into an opening where the stem should
be. For fixed angular momentum we find that no more than three different delay
times are possible. This induces striking nonperiodic structures in the delay
times that may be of importance for mesoscopic devices and should be accessible
to microwave experiments.
