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The quantum cat map is a model for a quantum system with underlying chaoticdynamics. In this paper we study the matrix elements of smooth observables inthis model, when taking arithmetic symmetries into account. We give explicitformulas for the matrix elements as certain exponential sums. With theseformulas we can show that there are sequences of eigenfunctions for which thematrix elements decay significantly slower then was previously conjectured. Wealso prove a limiting distribution for the fluctuation of the normalized matrixelements around their average.

On matrix elements for the quantized cat map modulo prime powers