Consider Birkhoff sums \(S_n = \sum_{k < n} v \circ T^k \) where \(T\) is a billiard with a flat cusp and stability index \(\alpha = \frac{3}{2}\), and \(v\) is an observable given by \( v(r, \varphi) = \cos(5 \varphi) \). Here \(\varphi\) is the post-collisional angle between particle's velocity and the normal to the boundary.

We choose a random initial condition and plot the process \(W_n(t)\), \(0 \leq t \leq 1\), given by \[ W_n(t) = \frac{1}{n^{2/3}} S_{\lfloor n t \rfloor} . \] We send \(n \to \infty\) and animate the graph. We also plot in light grey the graphs of \(t \mapsto c \, t^{2/3}\) with a few different values of \(c\).

Refresh the page to restart with a new initial contition.