APPROXIMATION BY ABSOLUTELY CONTINUOUS INVARIANT MEASURES OF ITERATED FUNCTION SYSTEMS WITH PLACE-DEPENDENT PROBABILITIES
Bibliographic record
Abstract
Let [Formula: see text] be the attractor (fractal) of a contractive iterated function system (IFS) with place-dependent probabilities. An IFS with place-dependent probabilities is a random map [Formula: see text] where the probabilities [Formula: see text] of switching from one transformation to another are functions of positions, that is, at each step, the random map [Formula: see text] moves the point [Formula: see text] to [Formula: see text] with probability [Formula: see text]. If the random map [Formula: see text] has a unique invariant measure [Formula: see text], then the support of [Formula: see text] is the attractor [Formula: see text]. For a bounded region [Formula: see text], we prove the existence of a sequence [Formula: see text] of IFSs with place-dependent probabilities whose invariant measures [Formula: see text] are absolutely continuous with respect to Lebesgue measure. Moreover, if [Formula: see text] is a compact metric space, we prove that [Formula: see text] converges weakly to [Formula: see text] as [Formula: see text] We present examples with computations.
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 | 0.002 | 0.010 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.003 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.003 | 0.003 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| 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".