On the Nonuniqueness of the Invariant Probability for I.I.D. Random Logisitc Maps
Bibliographic record
Abstract
Let $\{X_n\}^{\infty}_0$ be a Markov chain with values in $[0,1]$ generated by the iteration of random logistic maps defined by $X_{n+1}=f_{C_{n+1}}(X_n)\equiv C_{n+1}X_n(1-X_n)$, $n=0,1,2,\ldots\,$, with $\{C_n\}^{\infty}_1$ being independent and identically distributed random variables with values in $[0,4]$ and independent of $X_0$. This paper provides a class of examples where $C_i$ take only two values $\lambda$ and $\mu$ such that there exist two distinct invariant probability distributions $\pi_0$ and $\pi_1$ supported by the open interval $(0,1)$. This settles a question raised by R. N. Bhattacharya.
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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 | 0.009 | 0.044 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.003 | 0.001 |
| Science and technology studies | 0.003 | 0.012 |
| Scholarly communication | 0.004 | 0.006 |
| Open science | 0.003 | 0.004 |
| Research integrity | 0.003 | 0.005 |
| 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".