Pathology and asymmetry: centralizer rigidity for partially hyperbolic diffeomorphisms
Danijela Damjanovic, Amie Wilkinson, Disheng Xu
备注：
We significantly improve Theorems 1-4 in v2. The new statements now of Theorems 1&3 allow for arbitrary negatively curved surfaces, as well as much more general classes of geodesic flows in higher dimension. For Theorems 2&4, we classify the centralizer. We also eliminated the global rigidity results in the previous version. These results will appear in a forthcoming paper
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 $\mathbb Z^\ell$ or contains a smooth flow.
At the heart of this work are two very different rigidity phenomena. The first was discovered in [2,3] for a class of volume-preserving partially hyperbolic systems including those studied here, the disintegration of volume along the center foliation is either equivalent to Lebesgue or atomic. The second phenomenon is the rigidity associated to several commuting partially hyperbolic diffeomorphisms with very different hyperbolic behavior transverse to a common center foliation [25].
We introduce a variety of techniques in the study of higher rank, abelian partially hyperbolic actions: most importantly, we demonstrate a novel geometric approach to building new partially hyperbolic elements in hyperbolic Weyl chambers using Pesin theory and leafwise conjugacy, while we also treat measure rigidity for circle extensions of Anosov diffeomorphisms and apply normal form theory to upgrade regularity of the centralizer.
C1-generic conservative diffeomorphisms have trivial centralizer
本文证明了连通流形的C1辛同胚空间和C1保体积同胚空间都含有中心化子平凡的同胚剩余子集。(Les C1-génériques的保守性微分算子不是一个中心化的平凡算子。Nous montrons que l'espace des symplectomorphismes de classe C1 et l'espace des difféomorphismes de classe C1 préservant une form volume consiennent tous deux des sous-ensembles résiduels de difféomorphismes dont le centralisateur est trivial.）
