bibkey: gohlke2023measure authors: “P. Gohlke; A. Mitchell; D. Rust; T. Samuel” year: 2023 title: “Measure Theoretic Entropy of Random Substitution Subshifts” doi: “10.1007/s00023-022-01212-x” url: “https://link.springer.com/article/10.1007/s00023-022-01212-x” claim: “Example 5.3 gives the scalar entropy -(p log p + (1-p) log(1-p))/(2-p), maximised at the inverse golden ratio with value log of the golden ratio, for a different constant-length random substitution.” strata_touched: [] license: “citation-only” triage: “anchor”

Scalar entropy extremum supplier

The primary publisher article is in Annales Henri Poincaré 24, 277–323. Example 5.3 in §5 defines the random substitution

$$a\mapsto \{\begin{array}{ll}aa& \text{with probability }p,\\ ab& \text{with probability }1-p,\end{array}b\mapsto ba,0<p<1.$$

For its frequency measure ${\mu }_p$, the example uses Theorem 3.5 to give

$$h({\mu }_p)=-\frac{p\log p+(1-p)\log (1-p)}{2-p}.$$

It states that the maximum occurs at $p={\tau }^{-1}$, where $\tau$ is the golden ratio, and the value is $\log \tau \approx 0.481212$. This numerical value fixes the natural-log unit. In bits the scalar function is $h_2(p)/(2-p)$ and the maximum is ${\log }_2\tau$; the conversion divides the displayed entropy by $\ln 2$. The scalar extremum is literature-attested.

The single-lineage partition volume §7.4 consumes precisely this scalar precedent and proves its connection to the FIB geometry, including the strict equality condition. The deterministic rule $\alpha \mapsto \beta$, $\beta \mapsto ⟨\beta ,\alpha ⟩$ is not the random substitution of Example 5.3. Its uniform-root-position reference is not identified with the example’s frequency measure.

The example further gives a larger topological entropy and states that the golden frequency measure is not a measure of maximal entropy for that subshift. Neither its topological entropy nor a maximal-measure assertion is transferred to the partition model. The source is not Example 5.4’s random Fibonacci substitution.