Abstract:
Orbit separation dimension (OSD), also known as amorphic complexity,
is a relatively new invariant for almost automorphic dynamical systems.
We show that dynamical systems given by the translation action on spaces
of (almost automorphic) tilings generated by a primitive inflation
rule provide a rich class of examples for which the OSD is both
non-trivial and practically computable. For such tiling dymnamical
systems, Solomyak's overlap algorithm can be used to check that the
dynamical spectrum is pure-point, and it also provides as a by-product
all the ingredients that allow to determine (or at least estimate) the
OSD. We discuss all the necessary steps in this procedure, and then
illustrate the power of this invariant by numerous examples.