# Tower power for $S$-adics

## TL;DR

This paper revisits and clarifies results on $S$-adic systems in symbolic dynamics, introduces a minimal $S$-adic system with multiple ergodic measures, and provides a practical formula for computing cylinder measures.

## Contribution

It offers a standard language restatement of previous results, constructs a minimal $S$-adic system with multiple measures, and derives an efficient measure computation formula for non-primitive substitutions.

## Key findings

- Constructed a minimal $S$-adic system with $d$ ergodic measures.
- Developed a practical formula for cylinder measure computation.
- Presented detailed examples and model computations.

## 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.