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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\varepsilon >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. We study the semiclassical measures <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mu _{sc}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>μ</mi> <mrow> <mi mathvariant="italic">sc</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> of quasimodes of width <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\varepsilon \frac{\hbar }{|\log \hbar |}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mfrac> <mi>ħ</mi> <mrow> <mo stretchy="false">|</mo> <mo>log</mo> <mi>ħ</mi> <mo stretchy="false">|</mo> </mrow> </mfrac> </mrow> </math></EquationSource> </InlineEquation>, a critical-type regime when considering “delocalization". We derive a lower bound for the Kolmogorov-Sinai entropy of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mu _{sc}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>μ</mi> <mrow> <mi mathvariant="italic">sc</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> that depends explicitly on <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\varepsilon $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ε</mi> </math></EquationSource> </InlineEquation>, in the spirit of that given by Anantharaman-Koch-Nonnenmacher.

Entropy of Logarithmic Modes

期刊论文
发表日期：

2025年10月30日

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Suresh Eswarathasan

摘要

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 $\varepsilon >0$ $ε > 0$ . We study the semiclassical measures $\mu _{sc}$ $μ_{sc}$ of quasimodes of width $\varepsilon \frac{\hbar }{|\log \hbar |}$ $ε \frac{ħ}{| log ħ |}$ , a critical-type regime when considering “delocalization". We derive a lower bound for the Kolmogorov-Sinai entropy of $\mu _{sc}$ $μ_{sc}$ that depends explicitly on $\varepsilon $ $ε$ , in the spirit of that given by Anantharaman-Koch-Nonnenmacher.