LINEARIZATION OF COMPLEX HYPERBOLIC DULAC GERMS

The paper proves that a hyperbolic complex Dulac germ f(ζ)=ζ+β+o(1) on a standard quadratic domain RC admits a unique parabolic linearization ϕ solving ϕ∘f=ϕ+β on an f-invariant right subdomain, and that ϕ is itself a parabolic Dulac germ, with a real version when f is real. This is Theorem B and is established via: (i) existence and uniqueness by convergence of the Koenigs sequence (Theorem A) on an invariant right subdomain, and (ii) Dulac asymptotics by comparing ϕ to a formal Dulac linearization and solving a small homological equation with a telescoping series (Lemma 4.4) to control errors . The candidate’s solution instead constructs ϕ directly by solving the Abel cohomological equation with the telescoping sum H(ζ)=∑k≥0 r(f∘k(ζ)) (precisely the construction in Lemma 4.4 with h=−r), then proves uniqueness and Dulac asymptotics via a degree-by-degree finite-difference lemma and another telescoping correction, on an f-invariant right subdomain. This matches the paper’s logic for the homological step and differs only in how existence is obtained (telescoping versus Koenigs sequence). Both are valid under the same hypotheses on f and on the domain . Hence, both are correct, with different but compatible proofs.