By continuation from the hyperbolic limit of the cardioid billiard we show that there
isan abundance of bifurcationsin the family of lima¸con 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
mapsergo dicity occursfor more than one parameter the set of these parameter values
hasa complicated structure.
This is a pre-print. The definitive version: DULLIN, H.R. and BACKER, A., 2001. About ergodicity in the family of limacon billiards. Nonlinearity, 14(6), pp. 1673-1687, is available at: http://www.iop.org/EJ/journal/Non.