Symbol ratio minimax sequences in the lexicographic order

Abstract

Consider the space of sequences of k letters ordered lexicographically. We study the set M(α) of all maximal sequences for which the asymptotic proportions α of the letters are prescribed, where a sequence is said to be maximal if it is at least as great as all of its tails. The infimum of M(α) is called the α-infimax sequence, or the α-minimax sequence if the infimum is a minimum. We give an algorithm which yields all infimax sequences, and show that the infimax is not a minimax if and only if it is the α-infimax for every α in a simplex of dimension 1 or greater. These results have applications to the theory of rotation sets of beta-shifts and torus homeomorphisms.

26 pages. Corrected proof of Theorem 23(b) (previously Theorem 21(b)). Modified after discovering connections with work of Bruin and Troubetzkoy