p-adic geometry and dynamics
Bibliographic record
Abstract
Complex dynamics in a single variable is a well-studied field pioneered by Gaston Julia and Pierre Fatou.It deals with the iteration of rational maps on the Riemann sphere.Many beautiful fractal images arise such as the Mandelbrot set or the Julia sets of functions.As explored in [Sil07], dynamical systems has applications to arithmetic problems.The idea prompting this thesis was to look at the dynamics of p-adic correspondences and derive arithmetic information.We start our study of dynamics with the classical complex case.A quick diversion into nonarchimedean analysis is then needed before we move on to nonarchimedean dynamics, initially over P 1 (K) where K is a nonarchimedean field.Many results similar to those in the complex setting hold in this nonarchimedean situation.After illustrating some of these similarities, as well as some differences, we move on to build the theory of nonarchimedean geometry.Starting from the basics, we construct the category of rigid analytic spaces and give some important examples.These spaces, while interesting in their own right, are not true topological spaces and hence not apt spaces on which to study dynamics.In light of this, we construct Berkovich spaces (this time truly topological spaces) in the next section and give many examples.In our penultimate section, we look at how to define dynamics on P 1 Berk (K) by taking advantage of the tree structure.Finally, we explore some questions about dynamics of nonarchimedean correspondences, focusing specifically on the quadratic case.
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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 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.005 | 0.001 |
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".