Symbolic factors of -adic subshifts of finite alphabet rank

Abstract

AbstractThis paper studies several aspects of symbolic (i.e. subshift) factors of $\mathcal {S}$ -adic subshifts of finite alphabet rank. First, we address a problem raised by Donoso et al [Interplay between finite topological rank minimal Cantor systems, S-adic subshifts and their complexity. Trans. Amer. Math. Soc.374(5) (2021), 3453–3489] 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 on the previous results [B. Espinoza. On symbolic factors of S-adic subshifts of finite alphabet rank. Preprint, 2022, arXiv:2008.13689v2; N. Golestani and M. Hosseini. On topological rank of factors of Cantor minimal systems. Ergod. Th. & Dynam. Sys. doi:10.1017/etds.2021.62. Published online 8 June 2021]. 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 $\pi \colon (X,T)\to (Y,S)$ 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 [F. Durand. Linearly recurrent subshifts have a finite number of non-periodic subshift factors. Ergod. Th. & Dynam. Sys.20(4) (2000), 1061–1078].