On Matrix Elements for the Quantized Cat Map Modulo Prime Powers

The quantum cat map is a model for a quantum system with underlying chaotic dynamics. In this paper we study the matrix elements of smooth observables in this model, when taking arithmetic symmetries into account. We give explicit formulas for the matrix elements as certain exponential sums. With these formulas we can show that there are sequences of eigenfunctions for which the matrix elements decay significantly slower then was previously expected. We also prove a limiting distribution for the fluctuation of the normalized matrix elements around their average. Submitted: March 3, 2008., Accepted: August 11, 2008. This material is based upon work supported by the National Science Foundation under agreement No. DMS-0635607. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.