Forward limit sets of semigroups of substitutions

Ibai Aedo; Uwe Grimm; Ian Short

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arXiv · 2305.00078

Forward limit sets of semigroups of substitutions

Ibai Aedo·Uwe Grimm·Ian Short

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Abstract

We introduce the forward limit set $\Lambda$ 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 $\Lambda$ 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 images of all fixed points under $S$. It is usually difficult to compute a forward limit set explicitly; however, we show that, provided certain assumptions hold, $\Lambda$ is uncountable, and we supply upper bounds on its size in terms of logarithmic Hausdorff dimension.

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Ibai Aedo, Uwe Grimm, Ian Short. 2023-04-28. Forward limit sets of semigroups of substitutions. https://arxiv.org/abs/2305.00078

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