A family of chaotic billiards with variable mixing rates

We describe a one-parameter family of dispersing (hence hyperbolic, ergodic and mixing) billiards where the correlation function of the collision map decays as $1/n^a$ (here $n$ denotes the discrete time), in which the degree $a \in (1, \infty)$ changes continuously with the parameter of the family, $\beta$. We also derive an explicit relation between the degree $a$ and the family parameter $\beta$.