Sturmian Words and the Permutation that Orders Fractional Parts

Abstract

A Sturmian word is a map W from the natural numbers into {0,1} for which the set of {0,1}-vectors F_n(W):={(W(i),W(i+1),...,W(i+n-1))^T : i \ge 0} has cardinality exactly n+1 for each positive integer n. Our main result is that the volume of the simplex whose n+1 vertices are the n+1 points in F_n(W) does not depend on W. Our proof of this motivates studying algebraic properties of the permutation $\pi$ (depending on an irrational x and a positive integer n) that orders the fractional parts {1 x}, {2 x}, ..., {n x}, i.e., 0 < {\pi(1) x} < {\pi(2) x} < ... < {\pi(n) x} < 1. We give a formula for the sign of $\pi$, and prove that for every irrational x there are infinitely many n such that the order of $\pi$ (as an element of the symmetric group S_n) is less than n.