Limit Theorems for Random Non-Uniformly Expanding or Hyperbolic Maps

The paper proves a random complex RPF theorem for random Young towers via complex cone contraction and random compositions, culminating in (i)–(iii) of Theorem 2.4.5 (existence of analytic spectral data, positivity for real parameters, and exponential convergence with uniform constants) . It builds on a Lasota–Yorke inequality on weighted Lipschitz spaces , a carefully constructed random partition/minorization scheme and complex cone contraction for small complex perturbations of the transfer operators (Theorem 2.4.3) , together with perturbative estimates that control γ(L^z,k−L^0,k) at order |z| for k in a bounded window . By contrast, the model’s solution relies on a deterministic spectral projector/Dunford integral argument to extend real-parameter results to complex z and then asserts uniform decay of products of remainder operators R^z_{σ^{n-1}ω}⋯R^z_ω using only per-ω spectral radii. This step is not justified for random compositions and bypasses the essential complex-cone contraction machinery (and the use of general results for random compositions in [23]) that the paper explicitly deploys . Consequently, the model’s proof outline has a critical gap in establishing uniform exponential convergence for complex z in the random setting.