Existence and properties of equilibrium states of holomorphic endomorphisms of Pk

The paper establishes Theorem 1.2 exactly as stated: two explicit equivalent norms ‖·‖⋄1 and ‖·‖⋄2, uniform bounds sup_n‖λ^{-n}L^n‖⋄1 ≤ c together with ‖ρ‖⋄1, ‖1/ρ‖⋄1 ≤ c, and a spectral gap on the centered subspace with β < 1; moreover, for any 0 < δ < d^{γ/(2γ+2)} and A small, one can choose ‖·‖⋄2 so that β = 1/δ. This is proved via a pluripotential framework and new dynamical Sobolev-type spaces, not via inverse-branch geometry, culminating in Section 5 (Theorem 5.1) and Theorem 1.2 in the Introduction . By contrast, the candidate solution’s proof relies on (i) an asserted Doeblin minorization for the normalized operator T_φ and (ii) exponential decay of the “bad branch” weight; both are unjustified. In particular, T_φ^N(y,·) is supported on finitely many preimages, so a uniform Doeblin lower bound on every fixed-radius ball cannot hold, invalidating the claimed sup-norm contraction on mean-zero functions. The ‘bad branch mass decays exponentially’ claim is also not supported by the provided bounds and in fact the normalization estimates go the wrong way unless additional structure is proved. There is also circularity in invoking regularity of ρ to prove regularity of T_φ. Thus the model’s final statements match the paper’s theorem, but the proposed proof is flawed, while the paper’s proof is correct.