Uniform convergence 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 $\varphi$.

Statement recorded in the supplied source.

Can $\varphi^n(z)$ converge uniformly to $z_0$ on some nonempty open connected domain $D$ on which all these iterates are defined? Investigate also the additional hypotheses that $\varphi$ extends to an entire function, or that it is a germ of an algebraic function. [S2]

Short English statement

Can an irrationally indifferent holomorphic germ have iterates converging uniformly to its fixed point on a domain? Also consider germs extending to entire functions and algebraic germs.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 5300068 (AMR-052-0068). 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 2 and following paragraph. https://arxiv.org/abs/math/9205209v1.