Crystalline properties of eigenstates of quantum cat maps

Using the Bargmann-Husimi representation of quantum mechanics on a toroidal phase space, we study analytically the eigenstates of quantized cat maps. The linearity of the maps implies a close relationship between, on the one hand, classically invariant sublattices, on the other hand, the sets of Husimi zeros of certain quantum eigenstates. As a matter of fact, these zero patterns (or `constellations') do in some cases look like a crystal on the torus. As a consequence, we are able to build explicit families of eigenstates for which the zero patterns become uniformly distributed on the torus phase space in the limit $\hbar\to 0$. This result constitutes a first rigorous step towards the proof of semi-classical equidistribution of Husimi zeros for eigenstates of quantized one-dimensional chaotic systems.