Author’s abstract: Every \(\mathbb C^2\) action \(\alpha\) of \(\mathbb Z^k\), \(k\geq 2\), on the \((k+1)\)-dimensional torus whose elements are homotopic to the corresponding elements of an action \(\alpha_0\) by hyperbolic linear maps has exactly one invariant measure that projects to Lebesgue measure under the semiconjugacy between \(\alpha\) and \(\alpha_0\). This measure is absolutely continuous and the semiconjugacy provides a measure-theoretic isomorphism. The semiconjugacy has certain monotonicity properties and preimages of all points are connected. There are many periodic points for \(\alpha\) for which the eigenvalues for \(\alpha\) and \(\alpha_0\) coincide. We describe some nontrivial examples of actions of this type.