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Discussion and open questions

We could not yet give quantitative results about the topological entropy. This would be possible, if we could show the existence of a horseshoe.

The measurements of the metric entropy indicates that tex2html_wrap_inline389 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 [17]. For our tex2html_wrap_inline287 -billiards, we measured a local maximum around p=4, which corresponds to the real analytic billiard table given by the algebraic curve tex2html_wrap_inline983 . 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 tex2html_wrap_inline987 . Our experiments indicate that the linearly stable 2-periodic orbits are really stable and that the billiards tex2html_wrap_inline289 should therefore be not ergodic for all parameters p.

An interesting question is whether there exists a parameter value tex2html_wrap_inline993 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 tex2html_wrap_inline327 ) 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 tex2html_wrap_inline287 -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.

Oliver Knill, Jul 10 1998