On the fluctuations of matrix elements of the quantum cat map

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.