Ergodicity and mixing of noncommuting epimorphisms

We study mixing properties of epimorphisms of a compact connected finite-dimensional abelian group $X$. In particular, we show that a set $F$, $\mathrm{\mid }F\mathrm{\mid }>\dim X$, of epimorphisms of $X$ is mixing iff every subset of $F$ of cardinality $(\dim X)+1$ is mixing. We also construct examples of free nonabelian 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 nonabelian mixing subsemigroups.