Scarring of quasimodes on hyperbolic manifolds
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Abstract
Abstract Let N be a compact hyperbolic manifold, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:mi>M</mml:mi> <mml:mo>⊂</mml:mo> <mml:mi>N</mml:mi> </mml:mstyle> </mml:math> an embedded totally geodesic submanifold, and let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:mo>−</mml:mo> <mml:msup> <mml:mi>ħ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:msub> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:mi>N</mml:mi> </mml:msub> </mml:mstyle> </mml:math> be the semiclassical Laplace–Beltrami operator. For any <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:mi>ε</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mstyle> </mml:math> we explicitly construct families of quasimodes of energy width at most <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:mi>ε</mml:mi> <mml:mfrac> <mml:mi>ħ</mml:mi> <mml:mrow> <mml:mo stretchy="false">|</mml:mo> <mml:mi>log</mml:mi> <mml:mo></mml:mo> <mml:mi>ħ</mml:mi> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> </mml:mfrac> </mml:mstyle> </mml:math> which exhibit a ‘strong scar’ on M in that their microlocal lifts converge weakly to a probability measure which places positive weight on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:msup> <mml:mi>S</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> <mml:mi>M</mml:mi> </mml:mstyle> </mml:math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:mo stretchy="false">(</mml:mo> <mml:mo stretchy="false">↪</mml:mo> <mml:msup> <mml:mi>S</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> <mml:mi>N</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mstyle> </mml:math> . An immediate corollary is that any invariant measure on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:msup> <mml:mi>S</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> <mml:mi>N</mml:mi> </mml:mstyle> </mml:math> occurs in the ergodic decomposition of the semiclassical limit of certain quasimodes of width <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:mi>ε</mml:mi> <mml:mfrac> <mml:mi>ħ</mml:mi> <mml:mrow> <mml:mo stretchy="false">|</mml:mo> <mml:mi>log</mml:mi> <mml:mo></mml:mo> <mml:mi>ħ</mml:mi> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> </mml:mfrac> </mml:mstyle> </mml:math> .
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Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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