Research Problems in Function Theory — Problem 2.85

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.

Let $P_\lambda(z)=\lambda(z+z^2)$, where $|\lambda|=1$ and $\lambda$ is not a root of unity. There is a unique formal series $h_\lambda(z)=z+O(z^2)$ satisfying $P_\lambda(h_\lambda(z))=h_\lambda(\lambda z)$. Let $R(\lambda)\in[0,\infty]$ be its radius of convergence and put $m=\sup_\lambda R(\lambda)$. A periodic cycle of least period $q$ is repelling if $|(P_\lambda^{\circ q})'(z)|>1$ at a point of the cycle.

Statement recorded in the supplied source.

(a) Determine $m$ exactly, or approximate it with absolute error at most $10^{-10}$. (b) Is the supremum attained for some $\lambda=e^{2\pi i\alpha}$ with $\alpha$ a real quadratic irrational? (c) If $R(\lambda)=0$, does every disk $\{|z|<\varepsilon\}$, $\varepsilon>0$, contain an entire repelling periodic cycle of $P_\lambda$? [S2]

Short English statement

Find the maximal linearization radius for the quadratic family, determine whether a quadratic irrational rotation attains it, and ask whether failure of linearization forces repelling cycles arbitrarily close to the fixed point.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2302085 (AMR-022-2085). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory, arXiv:1809.07200v2 (2018), Problem 2.85. https://arxiv.org/abs/1809.07200.