  Periodic orbits are the central ingredients of modern semiclassical theories
and corrections to these are generally non-classical in origin. We show here
that for the class of generic polygonal billiards, the corrections are
predominantly classical in origin owing to the contributions from closed
almost-periodic (CAP) orbit families. Furthermore, CAP orbit families outnumber
periodic families but have comparable weights. They are hence indispensable for
semiclassical quantization.
