A Ruelle operator for holomorphic correspondences

The paper proves a Ruelle operator theorem for expansive holomorphic correspondences on X = supp(μ), including existence/simplicity of a positive eigenpair and uniform convergence of normalized iterates. Its strategy passes to a shift space of infinite forward orbits, establishes equicontinuity of normalized dual iterates, uses a density-of-preimages claim, and concludes uniform convergence; it also sketches a spectral-gap corollary for Hölder potentials. While the architecture is sound, two key ingredients are asserted with insufficient justification: (i) the density, in X, of backward images of any given point in X, which is used centrally in Lemma 10.1 and in the pressure/eigenvalue identification step; and (ii) compactness/convexity details of the cone Ω used with Schauder to obtain the eigenfunction. The candidate model solution is likewise strong in outline, giving a Walters–Lasota–Yorke approach with spectral gap and smoothing to Cα, but it assumes without proof that expansiveness on X yields a uniformly finite family of globally α-Lipschitz inverse branches (and that branching over X can be neglected), which is nontrivial for holomorphic correspondences. Therefore, both proofs have plausible cores but leave important steps unproven.