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 $v(r,\phi )=\cos (5\phi )$. Here $\phi$ 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 $W_n(t)$, $0\le t\le 1$, given by $$W_n(t)=\frac{1}{n^{2/3}}{S}_{\lfloor nt\rfloor }.$$ We send $n\to \mathrm{\infty }$ 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.