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Bibliographic record
Abstract
Abstract This paper deals with families of conformal iterated function systems (CIFSs). The space CIFS( X , I ) of all CIFSs, with common seed space X and alphabet I , is successively endowed with the topology of pointwise convergence and the so-called λ -topology. We show just how bad the topology of pointwise convergence is: although the Hausdorff dimension function is continuous on a dense G δ -set, it is also discontinuous on a dense subset of CIFS( X , I ). Moreover, all of the different types of systems (irregular, critically regular, etc.), have empty interior, have the whole space as boundary, and thus are dense in CIFS( X , I ), which goes against intuition and conception of a natural topology on CIFS( X , I ). We then prove how good the λ -topology is: Roy and Urbański [Regularity properties of Hausdorff dimension in infinite conformal IFSs. Ergod. Th. & Dynam. Sys. 25 (6) (2005), 1961–1983] have previously pointed out that the Hausdorff dimension function is then continuous everywhere on CIFS( X , I ). We go further in this paper. We show that (almost) all of the different types of systems have natural topological properties. We also show that, despite not being metrizable (as it does not satisfy the first axiom of countability), the λ -topology makes the space CIFS( X , I ) normal. Moreover, this space has no isolated points. We further prove that the conformal Gibbs measures and invariant Gibbs measures depend continuously on Φ∈CIFS( X , I ) and on the parameter t of the potential and pressure functions. However, we demonstrate that the coding map and the closure of the limit set are discontinuous on an important subset of CIFS( X , I ).
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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 | 0.001 |
| 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 it