Tobias Jäger (University of Jena)
We give an overview of some general concepts and methods for the
construction of irregular model sets, obtained via Meyer’s cut and
project schemes (CPS), with prescribed additional dynamical
properties. This allows to roduce specific examples at the very
opposite ends of dynamical complexity: On the one hand, one may obtain
irregular model sets with positive entropy [1]. On the other hand, it
is possible to construct model sets such that almost all fibres over
the torus parametrisation have a given finite cardinality k ∈ N [2,3].
Some intermeditate cases can be realised as well, for instance model
sets whose fibres are almost surely infinite, but which only allow a
finite number of ergodic invariant measures.
On the technical level, a special role is played by certain windows
with a very particular symmetry property, which we refer to as perfect
self-similarity (with respect to the window of the CPS) The
above-mentioned examples can all be realised in the Euklidean setting.
However, the methods can also be applied to the construction of
Toeplitz sequences or Toeplitz arrays over countable and residually
finite groups.
The presented results are contained in a series of works, including
joint work with Jamal Drewlo, Daniel Lenz, Gabriel Fuhrmann, Christian
Oertel, Maria Isabel Cortez and Jaime Gomez.
References:
[1] Model sets with positive entropy in Euclidean cut and project
schemes. T. Jäger, D. Lenz and C. Oertel. Ann. Sci. École Norm. Sup.
52:1073–1106, 2019.
[2] Irregular model sets and tame dynamics. G. Fuhrmann, E. Glasner,
T. Jäger and C. Oertel. Trans. Amer. Math. Soc. 374(5):3703-3734,2021.
[3] Toeplitz arrays obtained via cut and project schemes. M.I. Cortez,
J. Drewlo, J. Gomez and T. Jäger. In preparation.
[4] A survey on model sets. J. Drewlo, T. Jäger and D. Lenz. In preparation.