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: ifT : X → X is a mixing endomorphism of a compact metrizable abelian groupX, andT commutes with a completely positive entropyZ ²-actionS onX by continuous automorphisms, thenT has infinite entropy. Dedicated to the memory of Dr. Elizabeth Mary Hartley (1923–1998) The authors gratefully acknowledge support from EPSRC award no. 9570016X, N.S.F. grant No. DMS-94-01093, and the hospitality of the Warwick Mathematics Research Institute.