
## 3. Actions by rank one group case

1. ### 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?
1. Remark. [K. Vinhage] Examples exist where the centralizer is abelian and non-Lie. They are obtained by the Anosov-Katok 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 non-Lie 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 non-Kronecker 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 higher-rank groups, available at http://aimpl.org/rigidhigherrank.