Negative Lyapunov exponent of circle maps forced by expanding circle endomorphisms

The paper proves Theorem 1.1 by a counting argument on b-adic partitions, showing that for b large enough the fraction of “bad” levels along almost every orbit is ≤ 2(q/b) and hence below 1/(2(l+1)); combining this with a multiplicative estimate yields L(x,y) < (1/2) log C for a.e. (x,y). The key steps are Definition 1.1, Lemma 2.1 (counting lemma), Lemmas 2.2–2.3 (binomial tail + Borel–Cantelli), Lemma 3.1 (small slope of auxiliary graphs when b is large), and Lemma 4.1 (uniform bound on the number of bad children), culminating in Lemma 4.2 and Theorem 1.1. All these elements are explicit and consistent in the text . The candidate solution hinges on a false independence claim: it asserts that, under Lebesgue measure, x_j (determined by tail base-b digits) is independent of y_j (which in fact depends on the entire past base orbit and therefore on the same tail digits). This invalidates the key step E[I_j | past] ≤ ε and the subsequent martingale/Azuma argument. It also incorrectly claims that the result holds for all b ≥ 2 and that the “at most s intervals” hypothesis is unnecessary, both of which contradict the paper’s explicit reliance on a large b to create small distortion and on s to bound the number of bad subintervals per cylinder .