Entropy rigidity for 3D conservative Anosov flows and dispersing billiards
Bibliographic record
Abstract
Given an integer $k \geq 5$, and a $C^k$ Anosov flow $Φ$ on some compact connected $3$-manifold preserving a smooth volume, we show that the measure of maximal entropy (MME) is the volume measure if and only if $Φ$ is $C^{k-\varepsilon}$-conjugate to an algebraic flow, for $\varepsilon>0$ arbitrarily small. Besides the rigidity, we also study the entropy flexibility, and show that the metric entropy with respect to the volume measure and the topological entropy of suspension flows over Anosov diffeomorphisms on the $2$-torus achieve all possible values subject to natural normalizations. Moreover, in the case of dispersing billiards, we show that if the measure of maximal entropy is the volume measure, then the Birkhoff Normal Form of regular periodic orbits with a homoclinic intersection is linear.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.002 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".