Forward limit sets of semigroups of substitutions

Aedo, Ibai; Grimm, Uwe and Short, Ian (2023). Forward limit sets of semigroups of substitutions. arxiv/Forward limit sets of semigroups of substitutions (Early access).



We introduce the forward limit set Λ of a semigroup S generated by a family of substitutions of a finite alphabet, which typically coincides with the set of all possible s-adic limits of that family. We provide several alternative characterisations of the forward limit set. For instance, we prove that Λ is the unique maximal closed and strongly S-invariant subset of the space of all infinite words, and we prove that it is the closure of the set of all fixed points and their images under S. It is usually difficult to compute a forward limit set explicitly; however, we show that, provided certain assumptions hold, Λ is uncountable, and we supply upper bounds on its size in terms of logarithmic Hausdorff dimension.

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