Entropy bounds for endomorphisms commuting with \(k\) actions (Q1269756)

Let \(\Sigma=\{0,1,\dots,k-1\}^{\mathbb Z^2}\) be the full two-dimensional shift on \(k\) symbols, and let \(h: \Sigma\to\Sigma\) be a shift commuting homeomorphism on \(\Sigma\). \textit{M. Shereshevsky} [Indag. Math., New Ser. 4, 203--210 (1993; Zbl 0794.28010)] showed that \(h\) cannot be expansive and conjectured that \(h\) cannot have finite positive entropy. The authors formulate an algebraic analogue of this problem and prove the following special case of it. Let \(X\) be a compact metrizable abelian group. Let \(T: X\to X\) be a mixing endomorphism. Assume that \(T\) commutes with a \(\mathbb Z^2\)-action with completely positive entropy. Then \(T\) has infinite entropy.