  The dynamics of chaotic billiards is significantly influenced by coexisting
regions of regular motion. Here we investigate the prevalence of a different
fundamental structure, which is formed by marginally unstable periodic orbits
and stands apart from the regular regions. We show that these structures both
{\it exist} and {\it strongly influence} the dynamics of locally perturbed
billiards, which include a large class of widely studied systems. We
demonstrate the impact of these structures in the quantum regime using
microwave experiments in annular billiards.
