DOUBLY INTERMITTENT MAPS WITH CRITICAL POINTS, UNBOUNDED DERIVATIVES AND REGULARLY VARYING TAIL

The paper proves: (i) for every g in Ĝ (assumptions (A0)–(A2)) there is a unique (up to scale) σ-finite acim equivalent to Lebesgue via a double first-return inducing map G that is Gibbs–Markov and saturates I, and a lift (2.3) to the whole interval; (ii) a probability acim exists iff g ∈ G, where G is defined by the convergence of the two series ∑χ1_n, ∑χ2_n together with the balance condition |1/β1 − 1/β2| < 1. These are stated and proved via Proposition 2.2, Corollary 2.4, Proposition 2.6, and Theorem B, respectively . The candidate solution reaches the same final conclusions and employs a standard Gibbs–Markov inducing plus lift and Kac/Kakutani argument, with regular-variation asymptotics yielding χ1_n, χ2_n. The main discrepancy is methodological: the candidate introduces the balance condition |1/β1−1/β2|<1 as if needed to control distortion of the induced map, whereas the paper proves bounded distortion of the induced map G under (A0)–(A2) alone (Proposition 2.2; Sections 3.1–3.4) . This extra requirement in the candidate’s proof is unnecessary for Theorem A (existence of a σ-finite acim), though it is indeed part of the set G used in Theorem B . Aside from this, the candidate’s argument is broadly correct, albeit less sharp on tails (comparability vs. the paper’s precise asymptotics with explicit constants) .