self-intersecting topology makes folded chaotic WGMs amenable to the incorporation of waveguides because the openings have only small overlap with the waist of the mode. Although the field looks similar to Gaussian beam in the waist region, the latter are fundamentally different because they are always built upon stable periodic orbits, which in the simplest case requires a configuration corresponding to two focusing mirrors [@BabicBook1972]. In our design, the reflections are near-grazing along the entire convex part of the boundary, and there is no focusing-mirror configuration. Therefore, folded chaotic WGMs cannot be obtained in paraxial optics.

As an important correction to the ray limit in the presence of chaotic dynamics, [*dynamical localization*]{} has been invoked [@Nockel:1997lr] to explain the high Q factors of chaotic WGMs in convex cavities, and circular dielectrics with corrugated surface perturbations [@FangCao2005]. A hallmark of dynamical localization is an exponential decay of wave intensity as a function of a variable which in the classical ray picture exhibits diffusive time evolution. In corrugated perturbed circles, this variable is the angular momentum; and it remains a good quantity to characterize dynamical localization in chaotic WGMs of smooth but convex cavities [@Noeckel:T90-2001]. At each reflection, the angular momentum is 