Buff forms and invariant curves of near-parabolic maps

We prove a tameness result for the invariant curves of near-parabolic maps. In the special case of polynomials, this yields a necessary and sufficient condition for Hausdorff continuity of the external rays of a given periodic angle in terms of how the associated multipliers approach roots of unity. The main tool is the meromorphic 1-form introduced by X. Buff, which provides a convenient setup to study the local dynamics of near-parabolic maps in the complex plane. This is joint work with C. L. Petersen.