Dynamical quasitilings of amenable groups
Volume 66 / 2018
Bulletin Polish Acad. Sci. Math. 66 (2018), 45-55
MSC: Primary 37B05; Secondary 37B50.
DOI: 10.4064/ba8128-1-2018
Published online: 16 March 2018
Abstract
We prove that for any compact zero-dimensional metric space on which an infinite countable amenable group G acts freely by homeomorphisms, there exists a dynamical quasitiling with good covering, continuity, Følner and dynamical properties, i.e. to every x\in X we can assign a quasitiling \mathcal {T}_x of G (with all the \mathcal {T}_x using the same, finite set of shapes) such that the tiles of \mathcal {T}_x are disjoint, their union has arbitrarily high lower Banach density, all the shapes of \mathcal {T}_x are large subsets of an arbitrarily large Følner set, and if we consider \mathcal {T}_x to be an element of a shift space over a certain finite alphabet, then x \mapsto \mathcal {T}_x is a factor map.
