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arXiv · 2012.00715

On symbolic factors of $\mathcal{S}$-adic subshifts of finite topological rank

Bastián Espinoza
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Abstract

This paper studies several aspects of symbolic factors of $\mathcal{S}$-adic subshifts of finite alphabet rank. First, we address a problem raised in [DDPM20] about the topological rank of symbolic factors of $\mathcal{S}$-adic subshifts and prove that this rank is at most the one of the extension system, improving results from [E20] and [GH2020]. As a consequence of our methods, we prove that finite topological rank systems are coalescent. Second, we investigate the structure of fibers $\pi^{-1}(y)$ of factor maps $\pi\colon(X,T)\to(Y,T)$ between minimal $\mathcal{S}$-adic subshifts of finite alphabet rank and show that they have the same finite cardinality for all $y$ in a residual subset of $Y$. Finally, we prove that the number of symbolic factors (up to conjugacy) of a fixed subshift of finite topological rank is finite, thus extending Durand's similar theorem on linearly recurrent subshifts.

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Bastián Espinoza. 2020-12-01. On symbolic factors of $\mathcal{S}$-adic subshifts of finite topological rank. https://arxiv.org/abs/2012.00715

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