Symbol ratio minimax sequences in the lexicographic order

Abstract

Consider the space of sequences of k letters ordered lexicographically. We study the set M({\alpha}) of all maximal sequences for which the asymptotic proportions {\alpha} 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({\alpha}) is called the {\alpha}-infimax sequence, or the {\alpha}-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 {\alpha}-infimax for every {\alpha} in a simplex of dimension 1 or greater. These results have applications to the theory of rotation sets of beta-shifts and torus homeomorphisms.