How long is the chaotic boundary of a billiard

For two-dimensional quantum billiards we derive Weyl's law, i.e. the average density of states, for a subset of eigenstates concentrating on an invariant region $\Gamma$ of phase space. The leading term is proportional to the area of the billiard times the phase-space fraction of $\Gamma$. The boundary term is proportional to the fraction of the boundary where parallel trajectories belong to $\Gamma$. Our result is numerically confirmed for the mushroom billiard, where we determine the boundary lengths associated with chaotic and regular states, and for the elliptical billiard, where we consider rotating and oscillating states.