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This paper develops techniques which are used to answer a number of questions in the theory of equivalence relations generated by continuous actions of abelian groups. The methods center around the construction of certain specialized hyper-aperiodic elements, which produce compact subflows with useful properties. For example, we show that there is no continuous 3-coloring of the Cayley graph on F(2^Z^2), the free part of the shift action of Z^2 on 2^Z^2. With earlier work of the authors thisarXiv:1803.03872v1 fatcat:eugnyleahzgw7nlvy2da37tl5y