Continued fraction algorithm for Sturmian colorings of trees

Abstract

Factor complexity $b_{n}(\unicode[STIX]{x1D719})$ for a vertex coloring $\unicode[STIX]{x1D719}$ 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 $b_{n}(\unicode[STIX]{x1D719})=n+2$. 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.