Entropy bounds for endomorphisms commuting withK actions

Shereshevsky has shown that a shift-commuting homeomorphism from the two-dimensional full shift to itself cannot be expansive, and asked if such a homeomorphism can have finite positive entropy. We formulate an algebraic analogue of this problem, and answer it in a special case by proving the following: if T:X® X is a mixing endomorphism of a compact metrizable abelian group X, and T commutes with a completely positive entropy Z2-action S on X by continuous automorphisms, then T has infinite entropy.