On the quantum variance of matrix elements for the cat map on the 4-dimensional torus

For many classically chaotic systems, it is believed that in the semiclassical limit, the matrix elements of smooth observables approach the phase space average of the observable. In the approach to the limit the matrix elements can fluctuate around this average. Here we study the variance of these fluctuations, for the quantum cat map on $\mathbb{T}^4$. We show that for certain maps and observables, the variance has a different rate of decay, than is expected for generic chaotic systems.