Rigidity of higher rank abelian actions

Many different recent theorems in higher-dimensional dynamics are often described as instances of the phenomenon of rigidity. For instance, in measurable dynamics two Bernoulli automorphism are measurable conjugated if their entropy coincides. In this case a possible finer algebraic structure is neglected by the measurable conjugacy. Since there are many conjugacies, one describes this situation as being soft. However, in recent papers ([1], [2]) it has been shown that for some higher rank actions (where the individual automorphism are Bernoulli), every measurable conjugacy is in fact an algebraic one. Here there are only a few conjugacies, the systems are rigid. The aim of the project is to extend and refine those rigidity results of higher rank abelian actions: Topological and measurable rigidity will be considered for higher rank abelian actions on compact abelian groups, including the important case of toral automorphism, and extensions of this setting, for instance, to nilmanifolds. [1] A. Katok, S. Katok, K. Schmidt. Rigidity of measurable structure for Z^d-actions by automorphisms of a torus. preprint. [2] B. Kitchens, K. Schmidt. Isomorphism Rigidity of irreducible algebraic Z^d-actions. preprint.