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Discussion

Our numerical experiments provoke some open questions.

tex2html_wrap_inline1018 We were especially interested in the parameter value tex2html_wrap_inline756 , which is the supremum of all parameter values tex2html_wrap_inline406 for which tex2html_wrap_inline1024 . The numerical experiments do not decide clearly, if there exists a second caustics which is different from the canonical caustics tex2html_wrap_inline410 .

tex2html_wrap1060 Fig. 12. The same picture as Fig. 5. Some orbits for tex2html_wrap_inline498 . 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 tex2html_wrap_inline410 . On the same height to the right, there is an elliptic periodic orbit also above tex2html_wrap_inline410 . The last invariant curve around that island has a hexagonal shape.
tex2html_wrap_inline1018 It looks as if for tex2html_wrap_inline406 in the interior of a phase locking interval, there exists a neighborhood of the canonical invariant circle tex2html_wrap_inline410 , for which there exists no other invariant circle. One can argue that the stable or unstable manifold of the hyperbolic Birkhoff periodic orbits on the caustics prevent this if they do cross transversely outside the canonically invariant circle.

tex2html_wrap_inline1018 The curvature of the tables tex2html_wrap_inline400 has discontinuities. Hubacher's result [4] shows that near the boundaries of tex2html_wrap_inline440 , there exist no invariant curves. A result of Angenent [1] implies that for any tex2html_wrap_inline424 , the topological entropy tex2html_wrap_inline1050 of the billiard at tex2html_wrap_inline400 is positive. It would be interesting to get quantitative results about the topological entropy. Is it monotonically decreasing in tex2html_wrap_inline406 ? Is tex2html_wrap_inline1056 ?

tex2html_wrap1062 Fig. 13. Some orbits for tex2html_wrap_inline1058 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).
Next: References Up: Billiards that share a Previous: Aubry-Mather Sets and the
Oliver Knill, Jul 8 1998