Differentiable invariant manifolds of nilpotent parabolic points

The paper proves existence and regularity of one-dimensional invariant manifolds for two-dimensional maps with a nilpotent parabolic fixed point in reduced form F(x,y) = (x + c y, y + p(x) + y q(x) + u(x,y) + g(x,y)) and splits into three cases depending on k and l. It establishes explicit leading coefficients for the parameterization K/H and a polynomial internal dynamics R(t) = t + R_N t^N + R_{2N-1} t^{2N-1}, with R_N < 0, together with precise Cr/analytic regularity, and smallness/nonresonance conditions (notably the β-bound in case 2) that ensure a contraction in weighted spaces via the parameterization method and a posteriori arguments. These results and formulas are stated in Theorems 2.1 and 2.6 and supported by the construction in Section 3 (e.g., Proposition 3.1 and 3.4) and the fixed-point scheme (Sections 4–5) . The candidate solution mirrors the paper’s approach: it sets a normalizing jet matching through orders N and 2N−1 to fix the leading coefficients and R’s first two nonlinear terms, then solves the invariance equation as a fixed-point problem in weighted C^r spaces, invoking the sign/nonresonance conditions (including the β-bound) to control the linearized cohomological operator, and finally uses gauge freedom to make the internal dynamics polynomial of the stated form. The explicit leading-coefficient formulas and the case-by-case conditions coincide with those in the paper, and the contraction mechanism and a posteriori step align with the paper’s fiber-contraction framework and a posteriori theorems . Hence both are correct and substantially the same proof scheme.