## |

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
(0.5,0.5)) which
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 [4] shows that near the boundaries of , there exist no invariant curves. A result of Angenent [1] 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). |