Symbol ratio minimax sequences in the lexicographic order

TL;DR

This paper characterizes and computes the infimax sequences in lexicographic order for sequences with prescribed letter proportions, with implications for rotation sets in dynamical systems.

Contribution

It provides an algorithm to generate all infimax sequences and characterizes when the infimax is also a minimax, advancing understanding of sequence ordering and dynamical applications.

Findings

An algorithm for infimax sequences is developed.

Infimax sequences are not minimax iff they are the alpha-infimax for all alpha in a certain simplex.

Results have applications to rotation sets of beta-shifts and torus homeomorphisms.

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.