Ergodicity and mixing of non-commuting epimorphisms

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