Our numerical experiments provoke some open questions.
We were especially interested in the parameter value , which is the supremum of all parameter values for which . The numerical experiments do not decide clearly, if there exists a second caustics which is different from the canonical caustics .
Fig. 12. The same picture as Fig. 5. Some orbits for . This
time, we identified the dihedral symmetry of the phase space to see more
details. We see the parabolic periodic orbit (see around coordinates
is on the canonical invariant curve . On the same height to the right,
there is an elliptic periodic orbit also above .
The last invariant curve around that
island has a hexagonal shape.
The curvature of the tables has discontinuities. Hubacher's result  shows that near the boundaries of , there exist no invariant curves. A result of Angenent  implies that for any , the topological entropy of the billiard at is positive. It would be interesting to get quantitative results about the topological entropy. Is it monotonically decreasing in ? Is ?
|Fig. 13. Some orbits for for which the billiard table is very close to a triangle. Like in Fig. 12, we have identified the dihedral symmetry. (The arc length parameter displaying the first coordinate is slightly distorted since the program uses a convenient arc length instead of the real arc length).|