SLOW ENTROPY AND VARIATIONAL DYNAMICAL SYSTEMS

The paper proves: (i) Sturmian subshifts and Denjoy circle transformations are not variational at polynomial scale; (ii) a full Hausdorff-dimension subset of 3-IETs is strongly variational at polynomial scale, with h_top,aχ = h_μ,aχ = 1 for the polynomial family aχ(n)=n^χ (Theorems 1.2–1.3; proofs in §§5–6) . The candidate solution reaches the same classification and mechanisms (Sturmian/Denjoy: Kronecker ⇒ metric slow entropy zero; 3-IETs: Diophantine construction ⇒ metric exponent 1; idoc ⇒ linear topological growth). However, the model normalizes slow entropy via a ratio-of-logs definition and thus reports values as 1/χ instead of the paper’s convention “1” for the polynomial family; this is a definition/normalization mismatch (cf. paper’s Definitions §2) . The model also states the continuity-atom count for 3-IETs as 2n+1 (linear and harmless for the exponent), whereas the paper’s Proposition 6.5 shows |P_n| = d n in general, giving 3n for d=3 . After translating the model’s normalization to that of the paper, the conclusions and variational statements agree. The paper’s diophantine construction (badly approximable α with a one-parameter slice) yields a Hausdorff-dimension-2 parameter set and metric slow entropy 1 (Theorem 6.1 with Lemma 6.9, Proposition 6.10, Lemma 6.11) . For Sturmian and Denjoy, the paper uses measurable equivalence to rotations plus Ferenczi’s theorem to get metric slow entropy zero and computes the topological value 1 at polynomial scale (Theorem 5.3–5.4, Cor. 5.7, Theorem 4.2) .