Controlling strong scarring for quantized ergodic toral automorphisms

Abstract

We show that in the semiclassical limit the eigenfunctions of quantized ergodic symplectic toral automorphisms cannot concentrate in measure on closed orbits of the dynamics. More generally, we show that the mass of the pure point component of the limit measure 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.