Continued fraction algorithm for Sturmian colorings of trees

Abstract

Factor complexity b(n) (phi) for a vertex coloring phi of a regular tree is the number of classes of n-balls up to color-preserving automorphisms. Sturmian colorings are colorings of minimal unbounded factor complexity b(n) (phi) = n + 2. In this article, we prove an induction algorithm for Sturmian colorings using colored balls in a way analogous to the continued fraction algorithm for Sturmian words. Furthermore, we characterize Sturmian colorings in terms of the data appearing in the induction algorithm.