Every C2 action α of Zk, 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.
All Science Journal Classification (ASJC) codes
- Algebra and Number Theory
- Applied Mathematics