Rotational subsets of the circle under zd

This paper is a study of invariant sets that have “geometric” rotation numbers, which we call rotational sets, for the angle-tripling map , and more generally, the angle-d-tupling map for d⩾2. The precise number and location of rotational sets for σd is determined by d−1, -length open intervals, called holes, that govern, with some specifiable flexibility, the number and location of root gaps (complementary intervals of the rotational set of length ). In contrast to σ2, the proliferation of rotational sets with the same rotation number for σd, d>2, is elucidated by the existence of canonical operations allowing one to reduce σd to σd−1 and construct σd+1 from σd by, respectively, removing or inserting “wraps” of the covering map that, respectively, destroy or create/enlarge root gaps.