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Dynamical quasitilings of amenable groups

Downarowicz T., Huczek D.

Bulletin of the Polish Academy of Sciences. Mathematics

2018

Vol. 66, no. 1

45--55

We prove that for any compact zero-dimensional metric space X 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 ∈ X we can assign a quasitiling Tx of G (with all the Tx using the same, finite set of shapes) such that the tiles of Tx are disjoint, their union has arbitrarily high lower Banach density, all the shapes of Tx are large subsets of an arbitrarily large Følner set, and if we consider Tx to be an element of a shift space over a certain finite alphabet, then x ↦ Tx is a factor map.