Entropy of logarithmic modes

Abstract

Let $(M,g)$ be a compact, boundaryless, Riemannian manifold whose geodesic flow on its unit sphere bundle is Anosov. Consider the (semiclassical) Laplace-Beltrami operator on $M$. Let $\hat{I}\mu >0$. We study the semiclassical measures $\hat{I}{¼}_{sc}$ of quasimodes spectrally supported in intervals of width $\hat{I}\mu \frac{h}{|\log h|}$, a critical-type regime when considering ``delocalization". We derive a lower bound for the Kolmogorov-Sinai entropy of $\hat{I}{¼}_{sc}$ that depends explicitly on $\hat{I}\mu$, in the spirit of that given by Ananthamaran-Koch-Nonnenmacher.

Result is now general with an explicit dependence on $\hat{I}\mu$ established in the lower bound. 34 pages. All comments welcome