Buff forms and invariant curves of near-parabolic maps

Abstract

We introduce a general framework to study the local dynamics of near-parabolic maps using the meromorphic $1$-form introduced by X.~Buff. As a sample application of this setup, we prove the following tameness result on invariant curves of near-parabolic maps: Let $g(z)=\lambda z+O(z^2)$ have a non-degenerate parabolic fixed point at $0$ with multiplier $\lambda$ a primitive $q$th root of unity, and let $\gamma :]-\infty ,0]\to \mathbb{D}(0,r)$ be a $g^{\circ q}$-invariant curve landing at $0$ in the sense that $g^{\circ q}(\gamma (t))=\gamma (t+1)$ and ${\lim }_{t\to -\infty }\gamma (t)=0$. Take a sequence $g_n(z)={\lambda }_{n}z+O(z^2)$ with $|{\lambda }_n|\ne 1$ such that $g_n\to g$ uniformly on $\mathbb{D}(0,r)$ and suppose each $g_n$ admits a $g_n^{\circ q}$-invariant curve ${\gamma }_n:]-\infty ,0]\to \mathbb{C}$ such that ${\gamma }_n\to \gamma$ uniformly on the fundamental segment $[-1,0]$. If ${\lambda }_n^q\to 1$ non-tangentially, then ${\gamma }_n$ lands at a repelling periodic point near $0$, and ${\gamma }_n\to \gamma$ uniformly on $]-\infty ,0]$. In the special case of polynomial maps, this proves Hausdorff continuity of external rays of a given periodic angle when the associated multipliers approach a root of unity non-tangentially.

32 pages, 8 figures