Bibliographic record
Abstract
The family of $W-$shaped maps was introduced in 1982 by G. Keller. Based on the investigation of the properties of the maps, it was conjectured that instability of the absolutely continuous invariant measure (acim) can result only from the existence of small invariant neighbourhoods of the fixed critical point of the limiting map. We show that the conjecture is not true by constructing a family of $W-$shaped maps with acims supported on the entire interval, whose limiting dynamical behavior is described by a singular \nmeasure. We then generalize the above result by constructing families of $W-$shaped maps $\\{W_a\\}$ with a turning fixed point having slope $s_1$ on one side and $-s_2$ on the other. Each such $W_a$ map has an acim $\\mu_a$. Depending on whether $\\frac{1}{s_1}+\\frac{1}{s_2}$ is larger, equal, or smaller than 1, we show that the limit of $\\mu_a$ is a singular measure, a combination of singular and absolutely continuous measure or an acim, respectively. We also consider $W$-shaped maps satisfying $\\frac 1{s_1}+\\frac 1{s_2}=1$ and show that the eigenvalue $1$ of the associated Perron-Frobenius operator is not stable, which in turn implies the instability of the isolated spectrum. Motivated by the above results, we introduce the harmonic average of slopes condition, with which we obtain new explicit constants for the upper and lower bounds of the invariant probability density function associated with the map, as well as a bound for the speed of convergence to the density. Moreover, we prove stability results using Rychlik's Theorem together with the harmonic average of slopes condition for piecewise expanding maps.
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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 | 0.002 | 0.014 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.004 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.003 | 0.002 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.012 | 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".