Measure transfer and $S$-adic developments for subshifts

Abstract

Based on previous work of the authors, to any $S$-adic development of a subshift $X$ a "directive sequence" of commutative diagrams is associated, which consists at every level $n\ge 0$ of the measure cone and the letter frequency cone of the level subshift $X_n$ associated canonically to the given $S$-adic development. The issuing rich picture enables one to deduce results about $X$ with unexpected directness. For instance, we exhibit a large class of minimal subshifts with entropy zero that all have infinitely many ergodic probability measures. As a side result we also exhibit, for any integer $d\ge 2$, an $S$-adic development of a minimal, aperiodic, uniquely ergodic subshift $X$, where all level alphabets ${\mathcal{A}}_n$ have cardinality $d$, while none of the $d-2$ bottom level morphisms is recognizable in its level subshift $X_n\subset {\mathcal{A}}_n^{\mathbb{Z}}$.

A new section 3 has been added. The revised version is more self-contained.

References in corpus (1)

• Recognizability in S-adic shifts