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Measure transfer and S-adic developments for subshifts
- Part of
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- Journal:
- Ergodic Theory and Dynamical Systems / Volume 44 / Issue 11 / November 2024
- Published online by Cambridge University Press:
- 11 March 2024, pp. 3120-3154
- Print publication:
- November 2024
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- Article
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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 
$\mathcal A_n$ have cardinality 
$d,$ while none of the 
$d-2$ bottom level morphisms is recognizable in its level subshift 
$X_n \subseteq \mathcal A_n^{\mathbb {Z}}$.
