论文标题
$ \ mathbb r^d $和$ \ mathbb z^d $的动作的方向性和弱混合
Directional ergodicity and weak mixing for actions of $\mathbb R^d$ and $\mathbb Z^d$
论文作者
论文摘要
我们定义了方向$ l $ ergodicity的概念，较弱的混合和混合，以保存$ \ mathbb z^d $ action $ t $ t $在lebesgue概率空间$（x，μ）上，其中$ l \ subseteq \ subseteq \ mathbb r^d $是线性订阅。对于$ \ mathbb r^d $操作，这些概念清楚地与$ t $ to $ l $限制的相同属性相对应。对于$ \ mathbb z^d $ t $ t $，我们通过使用单位悬架$ \ wideTilde t $ to Direction $ l $以及$ l^2的子空间（\ widetilde x，\ wideTilde -μ）$ perpendicular垂直于悬架转移因子来定义它们。我们表明，对于$ \ mathbb z^d $操作，这些属性是光谱不变的，因为它们显然是用于$ \ Mathbb r^d $ Actions。我们表明，对于在两种情况下，对于弱混合动作$ t $，定向性的性质都意味着定向弱混合。对于Ergodic $ \ Mathbb z^d $ Actions $ t $，我们探讨了通过单位悬架定义的定向属性与$ \ Mathbb r^d $ Actions中的$ t $ empeddings之间的关系。还解决了通用问题和非凝聚和非狂热混合方向的结构。
We define notions of direction $L$ ergodicity, weak mixing, and mixing for a measure preserving $\mathbb Z^d$ action $T$ on a Lebesgue probability space $(X,μ)$, where $L\subseteq\mathbb R^d$ is a linear subspace. For $\mathbb R^d$ actions these notions clearly correspond to the same properties for the restriction of $T$ to $L$. For $\mathbb Z^d$ actions $T$ we define them by using the restriction of the unit suspension $\widetilde T$ to the direction $L$ and to the subspace of $L^2(\widetilde X,\widetilde μ)$ perpendicular to the suspension rotation factor. We show that for $\mathbb Z^d$ actions these properties are spectral invariants, as they clearly are for $\mathbb R^d$ actions. We show that for weak mixing actions $T$ in both cases, directional ergodicity implies directional weak mixing. For ergodic $\mathbb Z^d$ actions $T$ we explore the relationship between directional properties defined via unit suspensions and embeddings of $T$ in $\mathbb R^d$ actions. Genericity questions and the structure of non-ergodic and non-weakly mixing directions are also addressed.
