We could not yet give quantitative results about the topological
entropy. This would be possible, if we could show the existence of a
The measurements of the metric entropy indicates that
is piecewise convex with a discrete set of sharp minima. We do not know
the reason for this phenomenon but it reminds us of the breathing chaos
measured in the case of a billiard in a gravitational field .
For our -billiards, we measured a local maximum around
p=4, which corresponds to the real analytic
billiard table given by the algebraic curve .
The behaviour around of the entropy around p=2 is interesting.
We considered only linear stability and did not decide about the stability of
the elliptic periodic orbits. Such computations are involved and
have been done (computer assisted) in the case of Robnik billiard
Our experiments indicate that the linearly stable 2-periodic orbits are
and that the billiards should therefore be not ergodic
for all parameters p.
An interesting question is whether there exists a parameter value
with some homotopically nontrivial
invariant curve. KAM theory does not give them and Hubacher's result
excludes them only near the boundary of the phase space.
The numerical experiments
indicate however, that near p=1, there might be some. One must say however
that numerical computations near p=1 (and ) become doubtful due
to the fact that the curvature becomes huge at some places.
We did not determine the thength spectra of the tables nor the Dirichlet eigenvalues. Since the -unit ball is a natural convex region in the plane, it would be interesting to know, how the Dirichlet data behave in dependence of p.