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

A Haar component for quantum limits on locally symmetric spaces