INVARIANT MEASURES FOR RANDOM EXPANDING ON AVERAGE SAUSSOL MAPS

The paper proves, via a Lasota–Yorke inequality on the bounded-oscillation space V_α, that Cesàro averages H^s_ω of iterates h^k_ω are relatively compact and converge in L^1 to a random invariant density h_ω (Theorem 5.2), using the hybrid LY bound (4.9) and the explicit growth control Θ(ω) from (5.5) . It then shows r ≤ dim Y1(ω) for the number of ergodic skew-product ACIPs (Corollary 5.4) via an Oseledets splitting , and derives the packing bound r ≤ m(C)/(γ_N · ess inf_ω Θ(ω)^{N/α}) (Theorem 6.1), relying on the ‘positivity on a ball’ lemma for V_α (Lemma 2.5) . The candidate solution follows the same structure: (1) the same LY-driven uniform bounds and compactness argument (Kolmogorov–Riesz in place of [19, Lemma A.1]), (2) the same Oseledets-top-subspace argument and linear independence of ergodic densities, and (3) the same ball-positivity plus packing estimate producing the stated upper bound. Minor presentation differences include an explicit normalization remark for h_ω (the paper implicitly treats 1 as a seed while defining densities of mass 1) and different references for quasi-compactness; neither affects correctness. Overall, the proofs are essentially the same and correct.