Abstract
We explain and restate the results from our recent paper [2] 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 \(w \in \mathcal A^*\) and any invariant measure \(\mu \) on the subshift \(X_\sigma \) defined by any everywhere growing but not necessarily primitive or irreducible substitution \(\sigma : \mathcal A^* \rightarrow \mathcal A^*\). Several examples are considered in detail, and model computations are presented.
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Acknowledgements
The authors would like to thank Julien Cassaigne and Pascal Hubert for useful comments, as well as our marseillan symbolic dynamics community for its inspiring atmosphere. We would also like to thank the referee for his careful reading of the first version, and for having encouraged us to include Remark 4.6.
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Bédaride, N., Hilion, A. & Lustig, M. Tower power for S-adics. Math. Z. 297, 1853–1875 (2021). https://doi.org/10.1007/s00209-020-02582-w
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DOI: https://doi.org/10.1007/s00209-020-02582-w