We discover a rigidity phenomenon within the volume-preserving partially hyperbolic diffeomorphisms with 1-dimensional center. In particular, for smooth ergodic perturbations of certain algebraic systems—including the discretized geodesic flows over hyperbolic manifolds and certain toral automorphisms with simple spectrum and exactly one eigenvalue on the unit circle—the smooth centralizer is either virtually or contains a smooth flow.
At the heart of this work are two very different rigidity phenomena. The first was discovered by Avila, Viana, and the second author: for a class of volume-preserving partially hyperbolic systems including those studied here, the disintegration of volume along the center foliation is equivalent either to Lebesgue or atomic. The second phenomenon, described by the first and third authors, is the rigidity associated to several commuting partially hyperbolic diffeomorphisms with very different hyperbolic behavior transverse to a common center foliation.
We employ a variety of techniques, among them a novel geometric approach to building new partially hyperbolic elements in hyperbolic Weyl chambers using Pesin theory and leafwise conjugacy, measure rigidity via thermodynamic formalism for circle extensions of Anosov diffeomorphisms, partially hyperbolic Livšic theory, and nonstationary normal forms.
To the memory of Anatole Katok
"Pathology and asymmetry: Centralizer rigidity for partially hyperbolic diffeomorphisms." Duke Math. J. 170 (17) 3815 - 3890, 15 November 2021. https://doi.org/10.1215/00127094-2021-0053