Research-in-Teams Project:   Aperiodic order and complexity in substitution based systems

Substitution rules are a fundamental tool for modelling hierarchical and aperiodic structures. In particular, many mathematical models of quasicrystals—--such as the Penrose or the recently discovered Hat tiling --— are  generated by simple substitution schemes. These systems lack translational symmetry but exhibit long-range order,  reflected in the pure point spectrum of an associated dynamical system. A particularly well-behaved subclass in this context is formed by mean equicontinuous systems.
A classical problem is to distinguish such systems up to invariants, aiding both classification and comparison to physical structures.

S-adic systems have recently emerged as a powerful and unifying framework for studying symbolic systems built from substitution rules. They generalize classical substitutions by allowing a directive sequence of substitutions. By imposing conditions on this sequence, one can define various classes of symbolic systems, and for many of these, strong structural results have been established to aid in their analysis. This formalism encompasses both the symbolic systems modelling aperiodic order and the symbolic codings of IETs. The proposed project aims to develop combinatorial tools within the S-adic framework to study  rigidity, weak mixing, and a new promising topological invariant within these classes.

Dates of stay: March 9 -20, 2026 & April 27 - May 10, 2026

Research Team: Henk Bruin (U of Vienna, Austria), Bastián Espinoza (U of Liège, Belgium), Maik Gröger Jagiellonian U, Poland), Elzbieta Krawczyk (U of Vienna, Austria),