Symbol ratio minimax sequences in the lexicographic order

Abstract

Consider the space of sequences of $k$letters ordered lexicographically. We study the set ${\mathcal{M}}(\boldsymbol{{\it\alpha}})$of all maximal sequences for which the asymptotic proportions $\boldsymbol{{\it\alpha}}$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${\mathcal{M}}(\boldsymbol{{\it\alpha}})$is called the$\boldsymbol{{\it\alpha}}$-infimaxsequence, or the$\boldsymbol{{\it\alpha}}$-minimaxsequence if the infimum is a minimum. We give an algorithm which yields all infimax sequences, and show that the infimax isnota minimax if and only if it is the$\boldsymbol{{\it\alpha}}$-infimax for every$\boldsymbol{{\it\alpha}}$in a simplex of dimension 1 or greater. These results have applications to the theory of rotation sets of beta-shifts and torus homeomorphisms.