Dynamical quasitilings of amenable groups

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 $x\in X$ 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.