About ergodicity in the family of limaçon billiards (Q2758132)

This paper is devoted to the possibility of ergodicity in the family of billiards, which is given by the simplest nontrivial conformal mapping of the unit circle; in polar coordinates the boundary of the family of limaçon billiards is parametrized by NEWLINE\[NEWLINE\rho (\varphi)= 1+ \varepsilon \cos \varphi,\;\varphi\in [-\pi,\pi],\tag{1}NEWLINE\]NEWLINE where \(\varepsilon \in [0,1]\) is the family parameter. Note that this family of billiards is the relevant one in the context of quantum chaos. The authors 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.