Ergodicity in umbrella billiards

Abstract

We investigate a three-parameter family of billiard tables with circular arc boundaries. These umbrella-shaped billiards may be viewed as a generalization of two-parameter moon and asymmetric lemon billiards, in which the latter classes comprise instances where the new parameter is $0$. Like those two previously studied classes, for certain parameters umbrella billiards exhibit evidence of chaotic behavior despite failing to meet certain criteria for defocusing or dispersing, the two most well understood mechanisms for generating ergodicity and hyperbolicity. For some parameters corresponding to non-ergodic lemon and moon billiards, small increases in the new parameter transform elliptic $2$-periodic points into a cascade of higher order elliptic points. These may either stabilize or dissipate as the new parameter is increased. We characterize the periodic points and present evidence of new ergodic examples.