Discussion\[sec:Discussion\]
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Folded chaotic whispering-gallery modes in phase space
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Individual modes of the folded chaotic WGM type do not shift or broaden significantly when the attached waveguides are shortened into stubs. From this we conclude that leakage into the waveguides is not the limiting factor for the lifetimes of individual folded chaotic WGMs. However, this is not straightforward to reconcile with the ray-tracing results depicted in Figs. \[fig:Poincare-surface-of-section\] and \[fig:Two-self-intersecting-periodicOrbits\] (e):

A comparison between the numerically observed peak Q factors in Fig. \[fig:Wavenumber-sweeps\] and the classical ray escape into the waveguides provides a first indication that dynamical wave localization is essential in the formation of folded chaotic WGMs. We find that up to a set of measure zero (the trapped unstable periodic orbits), no matter where in the classical phase space we launch a ray, it escapes into the waveguide openings much too fast to explain the $Q$ factor found in the wave calculations.

The rapid ray escape can be viewed as a result of Mather’s theorem, because the non-convex billiard shape implies that the bounce dynamics does not constitute a twist map, and consequently neither the Lazutkin nor the Poincar[é]{}-Birkhoff theorems apply [@LazutkinBook]. What remains is a largely 