Rotational subsets of the circle under

This paper is a study of invariant sets that have “geometric” rotation numbers, which we call rotational sets, for the angle-tripling map σ 3 : T → T , and more generally, the angle-d-tupling map σ d : T → T for d ⩾ 2 . The precise number and location of rotational sets for σ d is determined by d − 1 , 1 d -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 ⩾ 1 d ). In contrast to σ 2 , the proliferation of rotational sets with the same rotation number for σ d , d > 2 , is elucidated by the existence of canonical operations allowing one to reduce σ d to σ d − 1 and construct σ d + 1 from σ d by, respectively, removing or inserting “wraps” of the covering map that, respectively, destroy or create/enlarge root gaps.