Forward limit sets of semigroups of substitutions

Abstract

We introduce the forward limit set $\Lambda$ of a semigroup $S$ generated by a family of substitutions of a finite alphabet, which typically coincides with the set of all possible s-adic limits of that family. We provide several alternative characterisations of the forward limit set. For instance, we prove that $\Lambda$ is the unique maximal closed and strongly $S$-invariant subset of the space of all infinite words, and we prove that it is the closure of the set of images of all fixed points under $S$. It is usually difficult to compute a forward limit set explicitly; however, we show that, provided certain assumptions hold, $\Lambda$ is uncountable, and we supply upper bounds on its size in terms of logarithmic Hausdorff dimension.