Uniqueness of large invariant measures for Z^k actions with Cartan homotopy data

Every C² action α of Z^k, k ≥ 2, on the (k+ 1)-dimensional torus whose elements are homotopic to the corresponding elements of an action α₀ by hyperbolic linear maps has exactly one invariant measure that projects to Lebesgue measure under the semiconjugacy between α and α₀. 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 α for which the eigenvalues for α and α₀ coincide. We describe some nontrivial examples of actions of this kind.