The entangled ergodic theorem in the almost periodic case

TL;DR

This paper proves the convergence of certain ergodic averages involving unitary operators in almost periodic systems, extending ergodic theorems to complex multiple correlation scenarios.

Contribution

It establishes the strong operator convergence of entangled ergodic averages for almost periodic unitary operators, generalizing previous results to multiple correlations.

Findings

Convergence of ergodic averages in almost periodic systems.

Application to multiple correlation Cesaro means.

Extension of ergodic theorems to entangled averages.

Abstract

Let $U$ be a unitary operator acting on the Hilbert space $\ch$, and $\a:\{1,..., 2k\}\mapsto\{1,..., k\}$ a pair partition. Then the ergodic average $$ \frac{1}{N^{k}}\sum_{n_{1},...,n_{k}=0}^{N-1} U^{n_{\a(1)}}A_{1}U^{n_{\a(2)}}... U^{n_{\a(2k-1)}}A_{2k-1}U^{n_{\a(2k)}} $$ converges in the strong operator topology provided $U$ is almost periodic, that is when $\ch$ is generated by the eigenvalues of $U$. We apply the present result to obtain the convergence of the Cesaro mean of several multiple correlations.