Symbolic factors of -adic subshifts of finite alphabet rank

Abstract This paper studies several aspects of symbolic (i.e. subshift) factors $\mathcal {S}$ -adic subshifts finite alphabet rank. First, we address a problem raised by Donoso et al [Interplay between topological rank minimal Cantor systems, S-adic and their complexity. Trans. Amer. Math. Soc. 374 (5) (2021), 3453–3489] about the prove that this is at most one extension system, improving on previous results [B. Espinoza. On Preprint , 2022, arXiv:2008.13689v2; N. Golestani M. Hosseini. systems. Ergod. Th. & Dynam. Sys. doi:10.1017/etds.2021.62. Published online 8 June 2021]. As consequence our methods, systems are coalescent. Second, investigate structure fibers $\pi ^{-1}(y)$ factor maps \colon (X,T)\to (Y,S)$ ${\mathcal S}$ show they have same cardinality for all y in residual subset Y . Finally, number (up to conjugacy) fixed subshift finite, thus extending Durand’s similar theorem linearly recurrent [F. Durand. Linearly non-periodic factors. 20 (4) (2000), 1061–1078].