New topics in ergodic theory

Abstract

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 $(\mathfrak{A},\alpha )$ be a $C^{\ast }$--dynamical system, where $\mathfrak{A}=K(H)$, and $\alpha =ad(U)$ is an automorphism implemented by the unitary $U$. We show that $$\underset{N\to +\infty }{\lim }\frac{1}{N}\sum _{n=0}^{N-1}{\alpha }^n=E,$$ pointwise in the weak topology of $\K(H)$. Here, $E$ is a conditional expectation projecting onto the $C^{\ast }$--subalgebra $$(\underset{z\in {\sigma }_{\mathrm{pp}}(U)}{⨁}{E}_{z}B(H)E_z)\bigcap K(H).$$ If in addition $U$ is weakly mixing with $\Omega \in H$ the unique up to a phase, invariant vector under $U$ and $\omega =<\cdot \Omega ,\Omega >$, we have the following recurrence result. If $A\in K(H)$ fulfils $\omega (A)>0$, and $0<m_1<m_2<...<m_l$ are natural numbers kept fixed, then there exists an $N_0$ such that $$\frac{1}{N}\sum _{n=0}^{N-1}\omega (A{\alpha }^{n{m}_1}(A){\alpha }^{n{m}_2}(A)...{\alpha }^{n{m}_l}(A))>0$$ for each $N>N_0$.

18 pages

Cited by in corpus (1)

• The entangled ergodic theorem and an ergodic theorem for quantum "diagonal measures"