Doubly Intermittent Full Branch Maps with Critical Points and Singularities

The paper proves Theorem E by (i) constructing a first-return induced Gibbs–Markov (GM) map G on Δ−0 via a double-inducing procedure, establishing full-branch, expansion, and bounded distortion properties (Proposition 2.2) and saturation ; (ii) obtaining sharp tails for the return-time and weighted visit counts τa,b (Proposition 2.6) and hence for τ=τ1,1 (Corollary 2.8) with exponents β1=k2ℓ1, β2=k1ℓ2, β=max{β1,β2} ; (iii) decomposing ϕ into endpoint values plus a Hölder remainder and proving L2/Dα properties for the induced observable Φ (Proposition 2.10; Corollary 2.11), then applying Gouëzel’s induced GM limit theorems to conclude CLT, √(n log n) CLT, or α-stable laws with α=1/βϕ under (H) or (H′) exactly as in Theorem E . The candidate solution follows the same blueprint: build a GM inducing scheme, compute τ-tails from local forms, decompose ϕ, verify (H)/(H′), apply GM limit theorems, and note σ2=0 iff ϕ is a coboundary. Minor differences are cosmetic: the paper fixes the base Δ−0 (plus a symmetric G+), while the candidate speaks of Δ0=Δ−0∪Δ+0; and the candidate sketches heuristic tail sizes for Φ̃ whereas the paper proves Lq and o(t−1/βϕ) bounds. These do not affect the conclusions. Overall, the model’s proof aligns tightly with the paper’s results and logic (see the statement and use of Theorem E and βϕ in ; return-time tails in ; induced observables and reduction steps in ).