Topological weak mixing and quasi-Bohr systems

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A minimal dynamical system (X, T) is called quasi-Bohr if it is a non-trivial equicontinuous extension of a proximal system. We show that if (X, T) is a minimal dynamical system which is not weakly mixing then some minimal proximal extension of (X, T) admits a nontrivial quasi-Bohr factor. (In terms of Ellis groups the corresponding statement is:AG′=G implies weak mixing.) The converse does not hold. In fact there are nontrivial quasi-Bohr systems which are weakly mixing of all orders. Our main tool in the proof is a theorem, of independent interest, which enhances the general structure theorem for minimal systems.