3. Actions by rank one group case

Centralizer of a diffeomorphism
Problem 3.1.
[K. Vinhage] Is there a $C^{\infty}$ diffeomorphism of a closed manifold which is transitive, but whose centralizer in $\textrm{Diff}^{\infty}(M)$ is not a finite dimensional Lie group/locally compact?
Remark. [K. Vinhage] Examples exist where the centralizer is abelian and nonLie. They are obtained by the AnosovKatok construction, and appear as closures of the iterates of the system. See, for instance, Theorem 3 in [arXiv:1708.02529]
There are two possible modifications:
 When the system is Anosov, it is known that the centralizer is discrete. Products can yield mixed behavior, so for general partially hyperbolic nonLie is possible. Is it true the accessible partially hyperbolic implies Lie centralizer?
 The only known constructions are by taking the closures of iterates for some rigid, but nonKronecker system. Are there others? In particular, are there examples of a transitive system whose centralizer is not a central extension of a Lie group?


Centralizer of a diffeomorphism
Problem 3.2.
[A. Wilkinson] Which ergodic affine maps $f:X\to X$ of a homogeneous space $X$ have the property that if $g$ is a perturbation of $f$ such that $Z_{\textrm{Diff}(X)}(g)$ and $Z_{\textrm{Diff(X)}}(f)$ are virtually isomorphic, then $g$ is topologically (smoothly) conjugate to an affine perturbation of $f$? Do all partially hyperbolic, ergodic affine transformations have this property?
Cite this as: AimPL: Global rigidity of actions by higherrank groups, available at http://aimpl.org/rigidhigherrank.