Controlling strong scarring for quantized ergodic toral automorphisms

We show that in the semi-classical limit the eigenfunctions of quantized ergodic symplectic toral automorphisms can not concentrate in measure on a finite number of closed orbits of the dynamics. More generally, we show that, if the pure point component of the limit measure has support on a finite number of such orbits, then the mass of this component must be smaller than two thirds of the total mass. The proofs use only the algebraic (i.e. not the number theoretic) properties of the toral automorphisms together with the exponential instability of the dynamics and therefore work in all dimensions.