Ergodic theory of affine isometric actions on Hilbert spaces
Ergodic theory of affine isometric actions on Hilbert spaces
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希尔伯特空间上仿射等距作用的遍历理论
DOI：
10.1007/s00039-021-00584-2
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发表时间：
2021
影响因子：
2.2
通讯作者：
Amine Marrakchi
中科院分区：
文献类型：
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作者：
Yuki Arano;Yusuke Isono;Amine Marrakchi
经典高斯函子与局部紧群 Ga 的每个正交表示相关联，保留 G 的概率测度的动作称为高斯动作。在本文中，我们通过关联希尔伯特空间上的每个仿射等距动作来概括这种构造，希尔伯特空间是非奇异高斯动作的单参数族，其遍历特性以非常微妙的方式与原始动作的几何形状相关。我们证明这些非奇异高斯行为表现出相变现象，并将其与仿射等距行为的新定量不变量联系起来。我们使用 Patterson-Sullivan 理论以及 Lyons-Pemantle 在树索引随机游走方面的工作，以便精确描述作用于树的群的仿射等距动作的相变。我们还证明，每个没有属性（T）的局部紧群都承认一个自由、弱混合且稳定类型的非奇异高斯分布。
The classical Gaussian functor associates to every orthogonal representation of a locally compact groupGa probability measure preserving action ofGcalled a Gaussian action. In this paper, we generalize this construction by associating to every affine isometric action ofGon a Hilbert space, a one-parameter family of nonsingular Gaussian actions whose ergodic properties are related in a very subtle way to the geometry of the original action. We show that these nonsingular Gaussian actions exhibit a phase transition phenomenon and we relate it to new quantitative invariants for affine isometric actions. We use the Patterson-Sullivan theory as well as Lyons-Pemantle work on tree-indexed random walks in order to give a precise description of this phase transition for affine isometric actions of groups acting on trees. We also show that every locally compact group without property (T) admits a nonsingular Gaussian that is free, weakly mixing and of stable type.