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 $ε>0$. We study the semiclassical measures $μ_{sc}$ of quasimodes spectrally supported in intervals of width $ε\frac{h}{|\log h|}$, a critical-type regime when considering ``delocalization". We derive a lower bound for the Kolmogorov-Sinai entropy of $μ_{sc}$ that depends explicitly on $ε$, in the spirit of that given by Ananthamaran-Koch-Nonnenmacher.

Theorems & Definitions (17)

• Remark 1.2.5
• Remark 1.2.6
• Remark 3.1.3
• proof : Proof of Lemma \ref{['lem:strong_micro']}
• Remark 3.1.4
• proof
• proof
• Remark 5.2.4
• proof
• proof