Density of periodic points, invariant measures and almost equicontinuous points of Cellular Automata
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Bibliographic record
Abstract
Revisiting the notion of µ-almost equicontinuous cellular automata introduced by R. Gilman, we show that the sequence of image measures of a shift ergodic measure µ by iterations of such automata converges in Cesaro mean to an invariant measure µc. If the initial measure µ is a Bernouilli measure, we prove that the Cesaro mean limit measure µc is shift mixing. Therefore we also show that for any shift ergodic and F-invariant measure µ, the existence of µ-almost equicontinuous points implies that the set of periodic points is dense in the topological support S(µ) of the invariant measure µ. Finally we give a non trivial example of a couple (µ-equicontinuous cellular automata F, shift ergodic and F-invariant measure µ) which has no equicontinuous point in S(µ).
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 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