About ergodicity in the family of limacon billiards

By continuation from the hyperbolic limit of the cardioid billiard we show that there is an abundance of bifurcations in the family of limacon billiar ds. The statistics of these bifurcation shows that the size of the stable i ntervals decreases with approximately the same rate as their number increas es with the period. In particular, we give numerical evidence that arbitrar ily close to the cardioid there are elliptic islands due to orbits created in saddle-node bifurcations. This shows explicitly that if in this one-para meter family of maps ergodicity occurs for more than one parameter, the set of these parameter values has a complicated structure.