Pathology and asymmetry: Centralizer rigidity for partially hyperbolic diffeomorphisms
Pathology and asymmetry: Centralizer rigidity for partially hyperbolic diffeomorphisms
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DOI：
10.1215/00127094-2021-0053
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发表时间：
2019-02
影响因子：
2.5
通讯作者：
Danijela Damjanović;A. Wilkinson;Disheng Xu
中科院分区：
文献类型：
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作者：
Danijela Damjanović;A. Wilkinson;Disheng Xu
在中心为1维的保体积部分双曲微分同态中发现了一种刚性现象。特别地，对于某些代数系统的光滑遍历扰动——包括至少3维的双曲流形上的离散测地流和单位圆上具有简单谱和恰好一个特征值的线性自同构——如果光滑中心化子的阿贝尔化具有足够高的秩，则该中心化子包含一个光滑的流。在维数$3$中，我们得到了一个全局二分类：对于一个保留可定向叶状圆的遍历时部分双曲微分同态$f$，扶正器实际上是平凡的，或者它包含一个光滑的流（在这种情况下，在有限覆盖范围内，$f$是一个保持面积的Anosov微分同态的光滑等距扩展）。这项工作的核心是两种非常不同的刚性现象。第一个现象是在[2,3]中发现的：对于一类保持体积的部分双曲型系统，包括本文所研究的系统，体积沿中心叶理的分解要么等同于勒贝格分解，要么等同于原子分解。另一种现象是与几个可交换部分双曲微分同态有关的刚性，它们具有非常不同的双曲行为，横向到一个共同的中心叶理[25]。我们在高阶阿贝尔部分双曲作用的研究中引入了各种技术：最重要的是，我们展示了一种新的几何方法，利用Pesin理论和叶共轭在双曲Weyl室中构建新的部分双曲单元，同时我们还处理了Anosov微分同态的圆扩展的测量刚性，并应用范式理论来提升扶正器的正则性。
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 of dimension at least 3 and linear toral automorphisms with simple spectrum and exactly one eigenvalue on the unit circle -- if the abelianization of the smooth centralizer has sufficiently high rank, then the centralizer contains a smooth flow. In dimension $3$, we obtain a global dichotomy: for an ergodic partially hyperbolic diffeomorphism $f$ that preserves an orientable foliation by circles, either the centralizer is virtually trivial, or it contains a smooth flow (in which case, up to a finite cover, $f$ is a smooth isometric extension of an area-preserving Anosov diffeomorphism). At the heart of this work are two very different rigidity phenomena. The first phenomenon 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 other 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.