Substitution invariant Sturmian words and binary trees

TL;DR

This paper explores substitution invariant Sturmian sequences, revealing their indexing by binary trees linked to Kepler's harmonic fractions, and extends results to inhomogeneous sequences.

Contribution

It introduces a novel indexing method for substitution invariant Sturmian sequences using binary trees related to Kepler's harmonic fractions.

Findings

Homogeneous substitution invariant Sturmian sequences indexed by binary trees.

Similar indexing results for inhomogeneous sequences.

Connection between Sturmian sequences and Kepler's harmonic fractions.

Abstract

We take a global view at substitution invariant Sturmian sequences. We show that homogeneous substitution invariant Sturmian sequences $s_{\alpha,\alpha}$ can be indexed by two binary trees, associated directly to Johannes Kepler's tree of harmonic fractions from 1619. We obtain similar results for the inhomogeneous sequences $s_{\alpha,1-\alpha}$ and $s_{\alpha,0}$.