Lyapunov exponents for transfer operator cocycles of metastable maps: A quarantine approach
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Bibliographic record
Abstract
This works investigates the Lyapunov–Oseledets spectrum of transfer operator cocycles associated to one-dimensional random <italic>paired tent maps</italic> depending on a parameter <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="epsilon"> <mml:semantics> <mml:mi> ε </mml:mi> <mml:annotation encoding="application/x-tex">\varepsilon</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , quantifying the strength of the <italic>leakage</italic> between two nearly invariant regions. We show that the system exhibits metastability, and identify the second Lyapunov exponent <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda 2 Superscript epsilon"> <mml:semantics> <mml:msubsup> <mml:mi> λ </mml:mi> <mml:mn>2</mml:mn> <mml:mi> ε </mml:mi> </mml:msubsup> <mml:annotation encoding="application/x-tex">\lambda _2^\varepsilon</mml:annotation> </mml:semantics> </mml:math> </inline-formula> within an error of order <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="epsilon squared StartAbsoluteValue log epsilon EndAbsoluteValue"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi> ε </mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>log</mml:mi> <mml:mo> </mml:mo> <mml:mi> ε </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\varepsilon ^2|\log \varepsilon |</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . This approximation agrees with the naive prediction provided by a time-dependent two-state Markov chain. Furthermore, it is shown that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda 1 Superscript epsilon Baseline equals 0"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi> λ </mml:mi> <mml:mn>1</mml:mn> <mml:mi> ε </mml:mi> </mml:msubsup> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\lambda _1^\varepsilon =0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda 2 Superscript epsilon"> <mml:semantics> <mml:msubsup> <mml:mi> λ </mml:mi> <mml:mn>2</mml:mn> <mml:mi> ε </mml:mi> </mml:msubsup> <mml:annotation encoding="application/x-tex">\lambda _2^\varepsilon</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are simple, and the only exceptional Lyapunov exponents of magnitude greater than <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="minus log 2 plus upper O left-parenthesis log log StartFraction 1 Over epsilon EndFraction slash log StartFraction 1 Over epsilon EndFraction right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo> − </mml:mo> <mml:mi>log</mml:mi> <mml:mo> </mml:mo> <mml:mn>2</mml:mn> <mml:mo>+</mml:mo> <mml:mi>O</mml:mi> <mml:mstyle scriptlevel="0"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo maxsize="1.623em" minsize="1.623em">(</mml:mo> </mml:mrow> </mml:mstyle> <mml:mi>log</mml:mi> <mml:mo> </mml:mo> <mml:mi>log</mml:mi> <mml:mo> </mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mi> ε </mml:mi> </mml:mfrac> <mml:mstyle scriptlevel="0"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo fence="true" stretchy="true" symmetric="true" maxsize="1.2em" minsize="1.2em">/</mml:mo> </mml:mrow> </mml:mstyle> <mml:mi>log</mml:mi> <mml:mo> </mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mi> ε </mml:mi> </mml:mfrac> <mml:mstyle scriptlevel="0"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo maxsize="1.623em" minsize="1.623em">)</mml:mo> </mml:mrow> </mml:mstyle> </mml:mrow> <mml:annotation encoding="application/x-tex">-\log 2+ O\Big (\log \log \frac 1\varepsilon \big /\log \frac 1\varepsilon \Big )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 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 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it