Parameterization of rotational sets under z^d
by
James Malaugh
University of Alabama at Birmingham

Much is known about rotational sets under z², the angle-doubling map on the unit circle. Specifically, that the rotational sets can be parameterized by semicircles of the circle (with parameter space [0, 1/2], representing one endpoint of the semicircle), and that a minimal rotational set must be either a Cantor set or a single periodic orbit. Moreover, the set of parameter values for which the rotation number is irrational is a set of Lebesgue measure 0. We identify an appropriate parameter space for rotational sets under z³, the angle-tripling map. We extend the results for z² to z³, with some fairly surprising revelations along the way. One of the extended results is that an appropriately defined rotation function is a measure theoretic two-dimensional Devil's staircase over the new parameter space. Many of the results also apply to z^d, d > 3.

Date received: February 28, 2003