Bibliographic record
Abstract
The renormalization theory of critical circle maps was developed in the late 1970's-early 1980's to explain the occurence of certain universality phenomena.These phenomena can be observed empirically in smooth families of circle homeomorphisms with one critical point, the so-called critical circle maps, and are analogous to Feigenbaum universality in the dynamics of unimodal maps.In the works of Ostlund et al. [ORSS] and Feigenbaum et al. [FKS] they were translated into hyperbolicity of a renormalization transformation.The first renormalization transformation in one-dimensional dynamics was constructed by Feigenbaum and independently by Coullet and Tresser in the setting of unimodal maps.The recent spectacular progress in the unimodal renormalization theory began with the seminal work of Sullivan [Sul1,Sul2,MvS].He introduced methods of holomorphic dynamics and Teichmüller theory into the subject, developed a quadratic-like renormalization theory, and demonstrated that renormalizations of unimodal maps of bounded combinatorial type converge to a horseshoe attractor.Subsequently, McMullen [McM2] used a different method to prove a stronger version of this result, establishing, in particular, that renormalizations converge to the attractor at a geometric rate.And finally, Lyubich [Lyu4,Lyu5] constructed the horseshoe for unbounded combinatorial types, and showed that it is uniformly hyperbolic, with one-dimensional unstable direction, thereby bringing the unimodal theory to a completion.The renormalization theory of circle maps has developed alongside the unimodal theory.The work of Sullivan was adapted to the subject by de Faria, who constructed in [dF1,dF2] the renormalization horseshoe for critical circle maps of bounded type.Later de Faria and de Melo [dFdM2] used McMullen's work to show that the convergence to the horseshoe is geometric.The author in [Ya1,Ya2] demonstrated the existence of the horseshoe for unbounded types, and studied the limiting situation arising when the combinatorial type of the renormalization grows without a bound.Despite the similarity in the development of the two renormalization theories up to this point, the question of hyperbolicity of the horseshoe attractor presents a notable difference.Let us recall without going into details the structure of the argument given by Lyubich in the unimodal case.The first part of Lyubich's work was to to endow the
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.000 | 0.001 |
| 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".