INTERMITTENCY GENERATED BY ATTRACTING AND WEAKLY REPELLING FIXED POINTS

The paper proves a phase transition for the existence of an absolutely continuous stationary (acs) probability for a random mixture of LSV-type maps S_α and maps R_{α,K}, with threshold η = Σ_{r∈Σ_R} p_r K_r^{-α_min}: no acs probability if η>1 (Theorem 1.1), and a unique acs probability if η<1 with a density that is decreasing, locally Lipschitz on each side of 1/2, bounded away from zero, and satisfying precise power upper bounds; moreover the skew product is ergodic (Theorem 1.2). These statements and definitions appear explicitly in the paper (η, γ; Theorems 1.1–1.2) and the proof strategy uses the annealed operator P=PF,p, an invariant function class (C0→C1→C2), and Arzelà–Ascoli to obtain a fixed point, then ergodicity to conclude uniqueness; nonexistence is shown via a Kac-type return-time argument (PF,p-operator in (3.10), invariance and power bounds via Lemma 3.4; Arzelà–Ascoli compactness Lemma 3.5; ergodicity/uniqueness; Kac’s Lemma in §2 and the return-time estimate in §3.1). These steps are documented in the cited portions of the text (PF,p and invariance, fixed point, ergodicity/uniqueness, local Lipschitz, and Kac’s lemma) . The candidate’s solution reproduces the same core mechanism: an invariant cone controlled by model weights φ_β(x)=x^{-α_min-1+β} on (0,1/2] and ψ_β(x)=(x-1/2)^{-1+β} on (1/2,1], and the key right-branch factor Σ_{r} p_r K_r^{-β}<1 for β∈(α_min,γ), yielding existence/regularity; uniqueness and ergodicity follow by standard arguments for the annealed operator; nonexistence for η>1 is derived via a return-time/renewal estimate equivalent to Kac’s lemma. Minor differences are stylistic (e.g., a projective-cone uniqueness argument in the model vs. an ergodicity-of-F argument in the paper) but the mathematical content and conclusions match. One caveat: the model speculates on the η=1 border by asserting a σ-finite acs measure; the paper does not claim this and even conjectures the absence of a probability in that case, requiring finer control near 1/2. This extra model remark is not needed for the main theorems and does not affect correctness of the solved parts.