遍历论的新课题
New topics in ergodic theory
摘要：
纠缠遍历定理涉及研究多重Cesaro平均$$\frac{1}{N^{k}}\sum_{n_{1},...,n_{k}=0}^{N-1} U^{n_{\a(1)}}A_{1}U^{n_{\a(2)}}... U^{n_{\a(2k-1)}}A_{2k-1}U^{n_{\a(2k)}} ,$$在强或弱算子拓扑中的收敛性，其中$U$是作用于希尔伯特空间$H$的酉算子，$\a:\{1,..., m\}\mapsto\{1,..., k\}$是$k$部分中$m$元素组成的集合的划分，最后$A_{1},...,A_{2k-1}$是作用于$H$的有界算子。在回顾纠缠遍历定理的最新结果的同时，我们提供了一些基于紧算子的动力系统的自然应用。也就是说，令$(\mathfrak A,α)$是一个$C^{*}$ -动力系统，其中$\mathfrak A=K(H)$和$α=ad(U)$是由酉$U$实现的自同构。证明了$$\lim_{N\to+\infty}\frac{1}{N}\sum_{n=0}^{N-1}α^{n}=E ,$$在$\K(H)$的弱拓扑中是点态的.这里，$E$是投射到$C^{*}$上的条件期望--子代数$$\bigg(\bigoplus_{z\inσ_{\mathop{\rm pp}}(U)} E_{z}B(H)E_{z}\bigg)\bigcap K(H) .$$如果另外$U$与$Ω\in H$弱混合，$U$和$ω=<\cdot Ω,Ω>$下唯一的直到相位的不变向量，我们有下面的递归结果。如果$A\in K(H)$满足$ω(A)>0$，并且$0<m_{1}<m_{2}<...<m_{l}$是保持固定的自然数，则存在一个$N_{0}$，使得对于每个$N>N_{0}$都有$$\frac{1}{N}\sum_{n=0}^{N-1}ω(Aα^{nm_{1}}(A)α^{nm_{2}}(A)... α^{nm_{l}}(A))>0$$。
英文摘要：
The entangled ergodic theorem concerns the study of the convergence in the strong, or merely weak operator topology, of the multiple Cesaro mean $$\frac{1}{N^{k}}\sum_{n_{1},...,n_{k}=0}^{N-1} U^{n_{\a(1)}}A_{1}U^{n_{\a(2)}}... U^{n_{\a(2k-1)}}A_{2k-1}U^{n_{\a(2k)}} ,$$ where $U$ is a unitary operator acting on the Hilbert space $H$, $\a:\{1,..., m\}\mapsto\{1,..., k\}$ is a partition of the set made of $m$ elements in $k$ parts, and finally $A_{1},...,A_{2k-1}$ are bounded operators acting on $H$. While reviewing recent results about the entangled ergodic theorem, we provide some natural applications to dynamical systems based on compact operators. Namely, let $(\mathfrak A,α)$ be a $C^{*}$--dynamical system, where $\mathfrak A=K(H)$, and $α=ad(U)$ is an automorphism implemented by the unitary $U$. We show that $$\lim_{N\to+\infty}\frac{1}{N}\sum_{n=0}^{N-1}α^{n}=E ,$$ pointwise in the weak topology of $\K(H)$. Here, $E$ is a conditional expectation projecting onto the $C^{*}$--subalgebra $$\bigg(\bigoplus_{z\inσ_{\mathop{\rm pp}}(U)} E_{z}B(H)E_{z}\bigg)\bigcap K(H) .$$ If in addition $U$ is weakly mixing with $Ω\in H$ the unique up to a phase, invariant vector under $U$ and $ω=<\cdot Ω,Ω>$, we have the following recurrence result. If $A\in K(H)$ fulfils $ω(A)>0$, and $0<m_{1}<m_{2}<...<m_{l}$ are natural numbers kept fixed, then there exists an $N_{0}$ such that $$\frac{1}{N}\sum_{n=0}^{N-1}ω(Aα^{nm_{1}}(A)α^{nm_{2}}(A)... α^{nm_{l}}(A))>0$$ for each $N>N_{0}$.
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