  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.
