Abstract:
In studying higher dimensional Schrödinger operators of quasicrystals, it is advantageous to find periodic approximations, given the known theory for crystals. Namely, we seek periodic operators such that their spectrum converges as a set to the spectrum of the limiting operator (of the quasi-crystal). To get such periodic approximations, one needs to study the convergence of the underlying dynamical systems.
We treat dynamical systems which are based upon substitutions. We find natural candidates of dynamical subsystems to approximate the substitution dynamical system. We provide a characterization when these converge and estimates on the rate of convergence.
Some well-known examples of 1-dimensional and 2-dimensional substitution systems are discussed during the talk. These results also apply to non-commutative substitution systems, recently introduced in [1]. This is based on a joint work with Ram Band, Siegfried Beckus and Felix Pogorzelski.
[1] Siegfried Beckus, Tobias Hartnick, and Felix Pogorzelski. Symbolic substitution systems
beyond abelian groups, 2021. Arxiv: 2109.15210
