On the dynamical Bogomolov conjecture for families of split rational maps
Bibliographic record
Abstract
We prove that Zhang’s dynamical Bogomolov conjecture holds uniformly along 1-parameter families of rational split maps and curves. This provides dynamical analogues of recent results of Dimitrov, Gao, and Habegger as well as Kühne. In fact, we prove a stronger Bogomolov-type result valid for families of split maps in the spirit of the relative Bogomolov conjecture. We thus provide first instances of a generalization of Baker-DeMarco’s conjecture to higher dimensions. Our proof contains both arithmetic and analytic ingredients. Our main analytic result may be viewed as a dynamical Ax–Lindemann-type theorem for split rational endomorphisms. More precisely, we show that weakly special curves under the action of a split map (f,g) of (PC1)2 are exactly those that lead to linear relations between the measures of maximal entropy of f and g. This extends a previous result of Levin and Przytycki. We further establish a height inequality for families of split maps and varieties comparing the values of a fiber-wise Call–Silverman canonical height with a height on the base and valid for most points of a non-preperiodic variety. This provides a dynamical generalization of a theorem of Haebegger and generalizes results of Call and Silverman as well as Baker to higher dimensions. In particular, we establish a geometric Bogomolov theorem for split rational maps and varieties of arbitrary dimension.
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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 | 0.001 | 0.004 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.002 | 0.002 |
| Scholarly communication | 0.001 | 0.003 |
| Open science | 0.000 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 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".