Transitive centralizer and fibered partially hyperbolic systems (Q6624408)

The emphasis here is on two examples: diffeomorphisms with a transitive centralizer, and perturbations of isometric extensions of Anosov diffeomorphisms of nilmanifolds. The geometric setting is a closed, connected smooth manifold \(M\). A centralizer \(\mathcal{Z}(f)\) of a diffeomorphism \(f: M \rightarrow M\) is the set of all diffeomorphisms that commute with \(f\) under composition. So all the iterates of \(f\) belong to \(\mathcal{Z}\) and form a normal subgroup. In a certain sense, the centralizer can be thought of as the set of smooth symmetries of \(f\). When \(f\) is a \(C^1\) generic diffeomorphism, the centralizer is trivial and consists of just the iterates of \(f\).\N\NIn the present paper and in [the authors, Duke Math. J. 170, No. 17, 3815--3890 (2021; Zbl 1497.37036)] the authors explore the question of what one can say about \(f\) if \(\mathcal{Z}(f)\) is nontrivial. Here they classify all smooth diffeomorphisms with a transitive centralizer. They turn out to be maps that preserve a principal fiber bundle structure, ones that act trivially on the base and minimally on the fibers. A primary result is that the centralizer of any partially hyperbolic diffeomorphism of a 3-dimensional non-toral nilmanifold is either virtually trivial, or that the diffeomorphism is an isometric extension of an Anosov diffeomorphism.