Measure transfer and S-adic developments for subshifts

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 ≥ 0 of the measure cone and the letter frequency cone of the level subshift Xₙ 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 ≥ 2 , an S -adic development of a minimal, aperiodic, uniquely ergodic subshift X , where all level alphabets Aₙ have cardinality $d,$ while none of the $d-2$ bottom level morphisms is recognizable in its level subshift Xₙ ⊆ AₙZ .