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.