1. Measure-theoretic mixing of finite topological rank systems

I construct the first examples of finite topological rank subshifts that are measure-theoretically mixing. These subshifts are uniquely ergodic topological models of a well-known family of mixing rank-one systems.

2. A characterization of linearly recurrent systems based on the structure of linear complexity subshifts

A theorem of F. Durand characterizes linearly recurrent subshifts as those generated by finitary, positive and proper directive sequences. In this note, I provide a similar result based on the S-adic structure of linear complexity subshifts that I recently obtained.

3. Worst-case number of non-recognizable levels for finite alphabet rank S-adic sequences

It was shown in [Béal-Perrin-Restivo-Steiner 2023] that an S-adic sequence with alphabet rank d has at most d-1 non-recognizable levels. In these notes, we build an S-adic sequence with alphabet rank d and d-2 non-recognizable levels. Our example is also primitive and generates a minimal, aperiodic and uniquely ergodic subshift at all levels. This complements the examples of [Bédaride-Hilion-Lustig 2022] and [Béal-Perrin-Restivo-Steiner 2023].