Exponential equidistribution of periodic points for endomorphisms of P^k

The paper proves that for any holomorphic endomorphism f: P^k -> P^k of algebraic degree d≥2, the periodic points of period n equidistribute toward the equilibrium measure µ at an exponential rate, uniformly over any Q_n with P_{n,γ} ⊂ Q_n ⊂ P_n, with speed ξ^{α n} independent of α (Theorem 1.1) . The proof is self-contained and hinges on: (i) a new sharp estimate on the mass of Monge–Ampère measures near algebraic sets (Proposition 3.1) ; (ii) a careful construction of many uniformly contracting inverse branches inside Manhattan cell decompositions, leading to exponentially many repelling periodic points in the small Julia set with multipliers bounded below (Section 5) ; and (iii) quantitative equidistribution for preimages (Theorem 4.5), used to control counts inside cells (Corollary 4.7) . The assembly of these ingredients yields the desired exponentially fast equidistribution and the exponentially small complement P_n \ P_{n,γ} (Section 6) . By contrast, the model’s solution relies on a central but incorrect step: it asserts an exponential convergence of the normalized pushforwards d^{-kn} (Id×f)^n_*[Δ] to π_2^*µ as (k,k)-currents. This is dimensionally inconsistent—π_2^*µ is a measure (2k,2k)-current, whereas [Γ_n]=(Id×f^n)_*[Δ] is a (k,k)-current—so the stated convergence cannot even be formulated as a pairing with smooth (k,k)-forms. Moreover, even after correcting the bidegree, the claimed spectral-gap-type convergence for these singular currents is not justified by the cited sources. The model also assumes an exponential sparsity of bad periodic points (those outside P_{n,γ}) from classical results; the paper shows that obtaining an exponential rate in this counting problem requires new input (notably Proposition 3.1 and the Manhattan inverse-branch scheme) rather than just the classical Briend–Duval/Yomdin technology. Finally, while the model acknowledges multiplicity issues and O(d^{(k-1)n}) corrections (indeed consistent with #P_n = d^{kn}+O(d^{(k-1)n}) ), the core analytic step and the control of the bad set are not established by its argument.