We study the fluctuations of the diagonal matrix elements of the quantum cat
map about their limit. We show that after suitable normalization, the fifth
centered moment for the Hecke basis vanishes in the semiclassical limit,
confirming in part a conjecture of Kurlberg and Rudnick.

We also study sums of matrix elements lying in short windows. For observables
with zero mean, the first moment of these sums is zero, and the variance was
determined by the author with Kurlberg and Rudnick. We show that if the window
is sufficiently small in terms of Planck's constant, the third moment vanishes
if we normalize so that the variance is of order one.