An orbit converging to an irrationally indifferent fixed point

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A holomorphic germ $\varphi$ at $z_0\in\mathbb C$ is a holomorphic map defined near $z_0$, with two representatives identified when they agree on a smaller neighborhood. Assume \[\varphi(z_0)=z_0,\qquad \varphi'(z_0)=e^{2\pi i\alpha},\qquad \alpha\in\mathbb R\setminus\mathbb Q.\] Iterates are considered wherever a representative and its analytic continuations make them defined. The domain sought below need not be invariant under $\varphi$.An orbit tending to $z_0$ here means a nonconstant forward orbit $z,\varphi(z),\varphi^2(z),\ldots$, defined for all forward times, with $\varphi^n(z)\ne z_0$ for every $n\geq0$ and $\varphi^n(z)\to z_0$.

Statement recorded in the supplied source.

Can such an orbit, never landing on $z_0$ at a finite time, converge to the irrationally indifferent fixed point $z_0$? [S2]

Short English statement

Can an orbit of an irrationally indifferent holomorphic germ approach its fixed point without ever landing on it?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 5300069 (AMR-052-0069). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Ben Bielefeld, Mikhail Lyubich and contributors, Problems in Holomorphic Dynamics, arXiv:math/9205209v1 (1992), Alexandre Eremenko and Mikhail Lyubich, "Wandering Domains for Holomorphic Maps, Question 3. https://arxiv.org/abs/math/9205209v1.