proportional to $\sin\chi$. However, an important common feature on which previous work relies is that the underlying classical dynamics exhibits a separation of time scales, without which diffusive behavior cannot be identified. Before $\sin\chi$ changes significantly, a whispering-gallery ray will have completed many round trips, allowing the azimuthal positions of boundary reflections to be averaged out [@Noeckel:T90-2001].

To identify wave localization in the folded WGMs to be discussed here, we have to follow a different route because angular momentum does not undergo the required slow diffusion. It jumps discontinuously because a figure-eight pattern entails periodic reversals of the sense of rotation. Therefore, we begin with a description of the ray dynamics in phase space before presenting the detailed results of our wave calculations in section \[sec:wavecalc\]. These two descriptions are then synthesized in section \[sec:Discussion\] to identify the classical structures on which the modes are localized, using Husimi projections of the numerical wave functions onto the ray phase space.

Cavity shape and ray phase space
================================

![(a-c): Three self-intersecting unstable periodic orbits with angles of incidence $\sin\chi\ge0.6$ in the billiard shape given by Eq.(\[eq:ShapeCurve\]) with $\epsilon=0.43$. The lengths $L_{1}$, $L_{2}$ of the orbits are slightly different. The same topology of 