About ergodicity in the family of limaçon billiards

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