Open Problems from the Workshop “emerging Applications of Measure Rigidity” Held at the American Institute of Mathematics

1.1. It is well-known that an Anosov diffeomorphism is structurally stable: every C-diffeomorphism which is sufficiently close in C-topology to an Anosov diffeomorphism is topologically conjugate to it. However, the conjugation map is not differentiable in general. On the other hand, Anosov actions by higher rank abelian groups exhibit much more rigid behavior (see [93] for the first result of this type). It was shown in [97] that most of known algebraic Anosov Zand Ractions, k ≥ 2, are locally C∞-rigid. Recall that a C∞-action of Z is called locally C∞-rigid if any C∞-action of Z which is sufficiently C-close to this action is conjugate to it by a C∞-map. A C∞-action of R is called locally C∞-rigid if any C-small perturbation of this action is C∞-conjugate to it up to an automorphism of R.