Parabolic dynamics according to Maurice Heins

TL;DR

This work reframes parabolic dynamics of unit-disk self-maps through Heins' left straightening, showing that a left straightening $h$ exists and is unique up to automorphisms, with the hyperbolic step given by $s^f(z)=\omega\bigl(h(z),h(f(z))\bigr)$. Using this tool, it proves that $s^f$ vanishes everywhere iff it vanishes at one point, enabling a streamlined Valiron-type analysis without full model theory. By transferring to the upper half-plane via the Cayley transform, it establishes a uniform Valiron limit $\frac{f^n-\tau_f}{f^n(z_0)-\tau_f}\to 1$ on compacta for parabolic maps, and shows that positive hyperbolic step implies tangential convergence to the Wolff point with a common slope. Overall, the paper provides a simple, invariant-distances-based route to resolve classic questions about parabolic holomorphic dynamics and clarifies the role of hyperbolic step in the asymptotic behavior.

Abstract

This paper, based upon an unpublished manuscript by Maurice Heins, answers a question posed by Valiron about the dynamics of parabolic self-maps of the unit disk in the complex plane, considerably simplifying arguments previously used for answering the same question. The main new tool introduced is the notion of left straightening of a sequence of iterates, that can be effectively employed for studying the hyperbolic step of a parabolic map.

Theorems & Definitions (24)

• Theorem 1
• Theorem 2
• proof
• Corollary 3
• proof
• Theorem 4
• proof
• Theorem 5
• Lemma 6
• proof