Log-scale equidistribution of zeros of quantum ergodic eigensections

Abstract

Under suitable hypotheses, a symplectic map can be quantized as a sequence of unitary operators acting on the $N$th powers of a positive line bundle over a Kähler manifold. We show that if the symplectic map has polynomial decay of correlations, then there exists a density one subsequence of eigensections whose masses and zeros become equidistributed in balls of logarithmically shrinking radii of lengths $|\log N{|}^{-\gamma }$ for some constant $\gamma >0$ independent of $N$.

The ergodicity assumption is relaxed from exponential to polynomial decay of correlations