Dynamics of Holomorphic Maps: Resurgence of Fatou coordinates, and Poly-time Computability of Julia Sets
Bibliographic record
Abstract
The present thesis is dedicated to two topics in Dynamics of Holomorphic maps. The first topic is dynamics of simple parabolic germs at the origin. The second topic is Polynomial-time Computability of Julia sets. Dynamics of simple parabolic germs. Let F be a germ with a simple parabolic fixed point at the origin: F( w) = w + w2 + O(w3). It is convenient to apply the change of coordinates z = −1/w and consider the germ at infinity [special characters omitted] The dynamics of a germ f can be described using Fatou coordinates. Fatou coordinates are analytic solutions of the equation &phis;( f(z)) = &phis;(z) + 1. This equation has a formal solution [special characters omitted]where ∑bjz–j is a divergent power series. Using Écalle's Resurgence Theory we show that [special characters omitted] can be interpreted as the asymptotic expansion of the Fatou coordinates at infinity. Moreover, the Fatou coordinates can be obtained from [special characters omitted] using Borel-Laplace summation. J. Écalle and S. Voronin independently constructed a complete set of invariants of analytic conjugacy classes of germs with a parabolic fixed point. We give a new proof of validity of Écalle's construction. Computability of Julia sets. Informally, a compact subset of the complex plane is called computable if it can be visualized on a computer screen with an arbitrarily high precision. One of the natural open questions of computational complexity of Julia sets is how large is the class of rational functions (in a sense of Lebesgue measure on the parameter space) whose Julia set can be computed in a polynomial time. The main result of Chapter II is the following: Theorem. Let f be a rational function of degree d ≥ 2. Assume that for each critical point c ∈ Jf the ω-limit set ω(c) does not contain either a critical point or a parabolic periodic point of f. Then the Julia set Jf is computable in a polynomial time.
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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 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| 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".