A Haar component for quantum limits on locally symmetric spaces

Abstract

We prove lower bounds for the entropy of limit measures associated to non-degenerate sequences of eigenfunctions on locally symmetric spaces of non-positive curvature. In the case of certain compact quotients of the space of positive definite $n\times n$ matrices (any quotient for $n=3$, quotients associated to inner forms in general), measure classification results then show that the limit measures must have a Lebesgue component. This is consistent with the conjecture that the limit measures are absolutely continuous.

References in corpus (2)

• Patterson-Sullivan distributions for symmetric spaces of the noncompact type
• Exponential decay for products of Fourier integral operators