Symbolic factors of $\mathcal{S}$-adic subshifts of finite alphabet rank

TL;DR

This paper investigates the structure and properties of symbolic factors of $ ext{S}$-adic subshifts with finite alphabet rank, establishing bounds on topological rank, fiber cardinalities, and finiteness of symbolic factors.

Contribution

It proves that the topological rank of symbolic factors is bounded by the extension system's rank, and shows finiteness results for symbolic factors and fiber cardinalities in these systems.

Findings

Topological rank of symbolic factors is at most that of the extension system.

Fibers of factor maps have constant finite cardinality on a residual set.

Number of symbolic factors up to conjugacy is finite for systems of finite topological rank.

Abstract

This paper studies several aspects of symbolic ({\em i.e.}\ subshift) 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 $\pi\colon(X,T)\to(Y,S)$ 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 [D00].