Ergodicity and mixing of non-commuting epimorphisms

We study mixing properties of epimorphisms of a compact connected finite-dimensional abelian group X. In particular, we show that a set F, with vertical bar F vertical bar > dim X, of epimorphisms of X is mixing if and only if every subset of F of cardinality (dim X) + 1 is mixing. We also construct examples of free non-abelian groups of automorphisms of tori which are mixing, but not mixing of order 3, and show that, under some irreducibility assumptions, ergodic groups of automorphisms contain mixing subgroups and free non-abelian mixing subsemigroups.