Tessellation and Lyubich–Minsky laminations associated with quadratic maps, II: Topological structures of 3-laminations
Bibliographic record
Abstract
According to an analogy to quasi-Fuchsian groups, we investigate the topological and combinatorial structures of Lyubich and Minsky’s affine and hyperbolic<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3"><mml:semantics><mml:mn>3</mml:mn><mml:annotation encoding="application/x-tex">3</mml:annotation></mml:semantics></mml:math></inline-formula>-laminations associated with hyperbolic and parabolic quadratic maps. We begin by showing that hyperbolic rational maps in the same hyperbolic component have quasi-isometrically the same<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3"><mml:semantics><mml:mn>3</mml:mn><mml:annotation encoding="application/x-tex">3</mml:annotation></mml:semantics></mml:math></inline-formula>-laminations. This gives a good reason to regard the main cardioid of the Mandelbrot set as an analogue of the Bers slices in the quasi-Fuchsian space. Then we describe the topological and combinatorial changes of laminations associated with hyperbolic-to-parabolic degenerations (and parabolic-to-hyperbolic bifurcations) of quadratic maps. For example, the differences between the structures of the quotient<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3"><mml:semantics><mml:mn>3</mml:mn><mml:annotation encoding="application/x-tex">3</mml:annotation></mml:semantics></mml:math></inline-formula>-laminations of Douady’s rabbit, the Cauliflower, and<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="z right-arrow from bar z squared"><mml:semantics><mml:mrow><mml:mi>z</mml:mi><mml:mo stretchy="false">↦<!-- ↦ --></mml:mo><mml:msup><mml:mi>z</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:annotation encoding="application/x-tex">z \mapsto z^2</mml:annotation></mml:semantics></mml:math></inline-formula>are described. The descriptions employ a new method of<italic>tessellation</italic>inside the filled Julia set introduced in Part I [<italic>Ergodic Theory Dynam. Systems</italic><bold>29</bold>(2009), no. 2] that works like external rays outside the Julia set.
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How this classification was reachedexpand
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 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".