TIME-RESTRICTED SENSITIVITY AND ENTROPY

The paper’s Theorem A states a_μ(x) = 1/h_μ(T,x) for μ-a.e. x and, in the ergodic case, T is μ-restricted sensitive iff h_μ(T)>0, aligning with the candidate’s claim. The paper’s route: define a_μ(x,δ) via first sensitive times, show 1/a_μ(x) = lim_{δ→0} inf_n[-log μ(B_n(x,δ))]/n (Lemma 3.5), and identify this limit with Brin–Katok local entropy (Section 2.2) to conclude Theorem A (proof immediately after Lemma 3.5). However, the proof of Lemma 3.4 contains a gap: it implicitly treats lim_{δ→0} μ(B_n(x,δ)) as μ({x}) for n≥1 (the line transitioning to “min_{1≤n≤N0} −log μ({x})/n”); in general B_n(x,δ)→∩_{i=0}^n T^{-i}({T^i x}) as δ→0, not {x}, unless injectivity or a similar property holds. This step (used to force μ({x})>0 and obtain a contradiction) needs repair; a corrected argument can bound the min by the n=0 term to reach the same contradiction but is not written as such in the text (see Lemma 3.4’s proof steps in the PDF). Aside from this fixable issue, the intended argument matches standard techniques and the main result is correct. The candidate solution is correct and gives a direct two-sided estimate using Brin–Katok: (i) for small ε, use liminf to get μ(B_n(x,ε)) ≤ e^{-(h(x)-η)n} for all large n and compare to μ(V) to get the upper bound a_μ(x) ≤ 1/h(x), and (ii) use limsup to produce Bowen balls of relatively large measure to block faster rates, giving a_μ(x) ≥ 1/h(x). The candidate should explicitly note that μ(V\B_n(x,ε)) ≥ μ(V)−μ(B_n(x,ε)) > 0 to match the paper’s positive-measure quantifier in the definition of restricted sensitivity. Overall: the theorem is right; the paper’s proof has a minor but real gap in Lemma 3.4; the model’s proof is sound when quantifier/measure details are made explicit. Citations: Theorem A statement is in the introduction (Theorem A) and proved after Lemma 3.5; key definitions and lemmas appear in Def. 3.1, Lemmas 3.2–3.5, with the Brin–Katok local entropy formula summarized in §2.2 (e.g., “for μ-a.e. x … the limsup and liminf agree”). See the arXiv PDF excerpts for these items (Theorem A and its proof, Lemma 3.5, Lemma 3.4 and its contentious step, and the definition of measure-theoretic restricted sensitivity). . Brin–Katok is summarized in §2.2 of the paper.