Tower power for $S$-adics

Abstract

We explain and restate the results from our recent paper arXiv:1503.08000.v3 in standard language for substitutions and $S$-adic systems in symbolic dynamics. We then produce as rather direct application an $S$-adic system (with finite set of substitutions $S$ on $d$ letters) that is minimal and has $d$ distinct ergodic probability measures. As second application we exhibit a formula that allows an efficient practical computation of the cylinder measure $\mu([w])$, for any word $w \in \cal A^*$ and any invariant measure $\mu$ on the subshift $X_\sigma$ defined by any everywhere growing but not necessarily primitive or irreducible substitution $\sigma: \cal A^* \to \cal A^*$. Several examples are considered in detail, and model computations are presented.