Monochromatic factorisations of words and periodicity

Abstract

In 2006 T. Brown asked the following question: Given a non-periodic infinite word $x=x_1{x}_2{x}_3\cdots$ with values in a non-empty set $\mathbb{A},$ does there exist a finite coloring $\phi :{\mathbb{A}}^{+}\to C$ relative to which $x$ does not admit a $\phi$-monochromatic factorisation, i.e., a factorisation of the form $x=u_1{u}_2{u}_3\cdots$ with $\phi (u_i)=\phi (u_j)$ for all $i,j\ge 1$? Various partial results in support of an affirmative answer to this question have appeared in the literature in recent years. In particular it is known that the question admits an affirmative answer for all non-uniformly recurrent words and various classes of uniformly recurrent words including Sturmian words. In this note we answer this question in general by showing that if $x=x_1{x}_2{x}_3\cdots$ is an infinite word with values in a non-empty set $\mathbb{A},$ then $x$ is periodic if and only if for every $2$-coloring $\phi :{\mathbb{A}}^{+}\to \{0,1\}$ there exists a $\phi$-monochromatic factorisation of $x.$ This characterization of periodicity of infinite words may be reformulated in the language of ultrafilters. Let $\beta {\mathbb{A}}^{+}$ denote the Stone-Cech compactification of the discrete semigroup ${\mathbb{A}}^{+}$ which we regard as the set of all ultrafilters on ${\mathbb{A}}^{+}.$ Then $x$ is periodic if and only if there exists $p\in \beta {\mathbb{A}}^{+}$ such that for each $A\in p$ there exists a factorisation $x=u_1{u}_2{u}_3\cdots$ with each $u_i\in A.$