New topics in ergodic theory

摘要: The entangled ergodic theorem concerns the study of convergence in strong, or merely weak operator topology, 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 acting on Hilbert space $H$, $\a:\{1,..., m\}\mapsto\{1,..., k\}$ partition set made $m$ elements $k$ parts, and finally $A_{1},...,A_{2k-1}$ are bounded operators $H$. While reviewing recent results about theorem, we provide some natural applications to dynamical systems based compact operators. Namely, let $(\mathfrak A,\alpha)$ be $C^{*}$--dynamical system, $\mathfrak A=K(H)$, $\alpha=ad(U)$ an automorphism implemented by $U$. We show that $$\lim_{N\to+\infty}\frac{1}{N}\sum_{n=0}^{N-1}\alpha^{n}=E pointwise topology $\K(H)$. Here, $E$ conditional expectation projecting onto $C^{*}$--subalgebra $$\bigg(\bigoplus_{z\in\sigma_{\mathop{\rm pp}}(U)} E_{z}B(H)E_{z}\bigg)\bigcap K(H) .$$ If addition weakly mixing with $\Omega\in H$ unique up phase, invariant vector under $\omega= $, have following recurrence result. $A\in K(H)$ fulfils $\omega(A)>0$, $0 0$$ for each $N>N_{0}$.