Entropy of semiclassical measures for nonpositively curved surfaces

TL;DR

This paper investigates the entropy properties of eigenfunctions on nonpositively curved surfaces, establishing a lower bound for the Kolmogorov-Sinai entropy of semiclassical measures related to the geodesic flow.

Contribution

It extends entropy bounds known for Anosov systems to nonpositively curved surfaces, highlighting key differences and similarities in the approach.

Findings

Kolmogorov-Sinai entropy is at least half of the Ruelle upper bound.

The main strategy parallels the Anosov case, with specific adaptations.

Provides insights into eigenfunction behavior on nonpositively curved manifolds.

Abstract

We study the asymptotic properties of eigenfunctions of the Laplacian in the case of a compact Riemannian surface of nonpositive sectional curvature. We show that the Kolmogorov-Sinai entropy of a semiclassical measure for the geodesic flow is bounded from below by half of the Ruelle upper bound. We follow the same main strategy as in the Anosov case (arXiv:0809.0230). We focus on the main differences and refer the reader to (arXiv:0809.0230) for the details of analogous lemmas.