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

TL;DR

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

Contribution

It proves that the topological rank of symbolic factors is bounded by the extension system's rank, shows fibers have uniform finite cardinality, and establishes finiteness of symbolic factors for finite topological rank systems.

Findings

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

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

Number of symbolic factors of a finite topological rank subshift is finite.

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.