On the maximal scarring for quantum cat map eigenstates

We consider the quantized hyperbolic automorphisms on the 2-dimensional torus (or generalized quantum cat maps), and study the localization properties of their eigenstates in phase space, in the semiclassical limit. We prove that if the semiclassical measure corresponding to a sequence of normalized eigenstates has a pure point component (phenomenon of ``strong scarring’'), then the weight of this component cannot be larger than the weight of the Lebesgue component, and therefore admits the sharp upper bound 1/2.