Rotational subsets of the circle under zd

AbstractThis paper is a study of invariant sets that have “geometric” rotation numbers, which we call rotational sets, for the angle-tripling map σ3:T→T, and more generally, the angle-d-tupling map σd:T→T for d⩾2. The precise number and location of rotational sets for σd is determined by d−1, 1d-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 ⩾1d). 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