Natural measures and statistical properties of non-statistical maps with multiple neutral fixed points

The paper proves Theorem A (for α∈(0,1), H1–H2) that L(x)=S almost everywhere cleanly via a distributional approach: first, Lemma 3.2 gives the easy inclusion L(x)⊂S, using that the complement of small neighborhoods of the fixed points has finite µ-measure under H1; then Theorem B shows en ⇒ νZα,p̄, and Lemma 3.6 establishes full support of νZα,p̄ when α∈(0,1); a standard ergodic argument then forces every ν∈S to appear as a limit point, hence L(x)=S (proof of Theorem A) . The abstract setup and assumptions H1–H2 are stated explicitly, and Theorem A is clearly formulated . By contrast, the candidate’s block/Borel–Cantelli construction hinges on unproven quasi-independence estimates for patterns of long excursions and on applying a dependent second Borel–Cantelli lemma to indicator events that are not shown to be in the mixing class required by the Gibbs–Markov spectral-gap framework; the asserted product-type lower bounds for multi-step cylinder events and the PD(α,0)-style heuristics are not justified under H1–H2 alone. Hence, while the candidate’s conclusion matches Theorem A, the argument as written has substantial gaps that the paper’s proof avoids by appealing to the multiray arcsine law and distributional convergence .