论文标题
非语音高斯动作：超越混合案例
Nonsingular Gaussian actions: beyond the mixing case
论文作者
论文摘要
在真正的希尔伯特空间上，每一个$ g $的仿射等距动作$α$都会在相关的高斯概率空间上产生非义动作的$ \hatα$ $ g $。在最近的论文[AIM19]中，当基础正交表示$ g $的基础正交表示$π$时，就建立了有关这些动作的恐怖和克里格类型的几个结果。当$π$仅是微弱混合时，我们开发了证明过时性的新方法。我们确定了全部的$ \hatα$的类型。使用Cantor措施，我们给出了III类型$ _1 $ ERGODIC高斯动作的示例，其基础表示不混合，甚至具有Dirichlet措施作为光谱类型。我们还为高斯偏斜产品动作提供了非常普遍的遗传性结果。
Every affine isometric action $α$ of a group $G$ on a real Hilbert space gives rise to a nonsingular action $\hatα$ of $G$ on the associated Gaussian probability space. In the recent paper [AIM19], several results on the ergodicity and Krieger type of these actions were established when the underlying orthogonal representation $π$ of $G$ is mixing. We develop new methods to prove ergodicity when $π$ is only weakly mixing. We determine the type of $\hatα$ in full generality. Using Cantor measures, we give examples of type III$_1$ ergodic Gaussian actions of $\mathbb{Z}$ whose underlying representation is non mixing, and even has a Dirichlet measure as spectral type. We also provide very general ergodicity results for Gaussian skew product actions.
