Ela Krawczyk (Jagiellonian University, Krakow):
Title: Amorphic complexity, tameness, and nullness of substitution shifts.
Abstract: Amorphic complexity---introduced by Fuhrmann, Gröger, and Jäger---is a relatively new invariant of topological dynamical systems useful in the study of aperiodic
order (i.e. mathematical models of quasicrystals) and low complexity dynamics. Roughly speaking, finiteness of amorphic complexity corresponds to systems with discrete
spectrum and continuous eigenvalues.
We study amorphic complexity in the class of automatic systems--- symbolic systems arising from constant length substitutions. We provide a closed formula for the amorphic
complexity of any automatic system and show that tameness/nullness of such systems can be succinctly characterized through amorphic complexity: An infinite minimal automatic
system is tame if and only if it is null if and only if its amorphic complexity is one.
Our proof uses methods from fractal geometry and introduces some new dynamically-defined pseudometrics. These methods seem suitable for study of other symbolic systems of
S-adic nature including nonconstant length substitutions and Toeplitz subshifts. Time permitting we will touch on some possible generalisations in these directions.
The talk is based on a joint work with Maik Gröger.