Measure transfer and $S$-adic developments for subshifts

TL;DR

This paper introduces a framework for analyzing subshifts using $S$-adic developments and directive sequences, revealing new insights into measure properties and ergodic measures of minimal subshifts.

Contribution

It develops a novel approach linking $S$-adic structures with measure cones, enabling direct deductions about subshift properties and constructing examples with specific ergodic characteristics.

Findings

Identifies a large class of minimal zero-entropy subshifts with infinitely many ergodic measures.

Constructs $S$-adic developments of minimal, aperiodic, uniquely ergodic subshifts with fixed alphabet size.

Shows that certain morphisms are not recognizable in their level 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 $n \geq 0$ 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 $d \geq 2$, an $S$-adic development of a minimal, aperiodic, uniquely ergodic subshift $X$, where all level alphabets ${\cal A}_n$ have cardinality $d\,$, while none of the $d-2$ bottom level morphisms is recognizable in its level subshift $X_n \subset {\cal A}_n^\mathbb Z$.