THE ONE-HOLE TO 2-HOLE TRANSITION FOR CANTORI

The gaps in a cantorus come in orbits, which we call ''holes''. In the space of parameters (a, b) for the ''two-harmonic'' reversible area-preserving twist map family, y' = y - a/2pi sin 2pix - b/4pi sin 4pix, x' = x + y' (mod 1) , application of the idea of the anti-integrable limit establishes that there must-be one to two-hole transitions for cantori of all irrational rotation numbers. We have numerically located a curve in parameter space across which a one-hole cantorus of golden rotation number develops a second hole, and we present results on scaling behaviour of several quantities near this interesting transition.