UNIQUENESS OF LARGE INVARIANT MEASURES FOR Z^k ACTIONS WITH CARTAN HOMOTOPY DATA

Every C² action ® of Z^k, k ≥ 2, on the (k +1)-dimensional torus whose elements are homotopic to the corresponding elements of an action ®0 by hyperbolic linear maps has exactly one invariant measure that projects to Lebesguemeasure under the semiconjugacy between ® and ®0. Thismeasure is absolutely continuous and the semiconjugacy provides ameasure-theoretic isomorphism. The semiconjugacy has certain monotonicity properties and preimages of all points are connected. There are many periodic points for ® for which the eigenvalues for ® and ®0 coincide. We describe some nontrivial examples of actions of this kind.