Research Problems in Function Theory — Problem 2.84

Definitions and mathematical statement

For $\lambda\in\mathbb C$ with $|\lambda|=1$, let $P_\lambda(z)=\lambda(z+z^2)$. Assume $\lambda$ is not a root of unity, equivalently $\lambda=e^{2\pi i\alpha}$ for some irrational real $\alpha$. The point zero is an irrationally indifferent fixed point. The unique finite critical point is $c=-1/2$, because $P_\lambda'(z)=\lambda(1+2z)$. Let $J(P_\lambda)$ be the Julia set, defined as the complement in the sphere of the open set where the iterates $\{P_\lambda^n:n\ge0\}$ are normal in the spherical metric. Does there exist such a parameter $\lambda$ for which the positive critical orbit is dense in that Julia set? \[ \overline{\{P_\lambda^n(-1/2):n\ge1\}}=J(P_\lambda). \] The closure is in the complex plane, or equivalently in the sphere since a polynomial Julia set is compact and bounded. The assertion asks for one suitable irrational unit-modulus parameter, not for every such parameter. Merely accumulating on part of the Julia set, or having a dense orbit for some point other than the critical point, is insufficient. No assumption of linearizability at zero is added to the parameter requirement.

Short English statement

Is there an irrational unit-modulus quadratic parameter for which the critical orbit is dense in the entire Julia set?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2302084 (AMR-022-2084), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Hayman - Research Problems in Function Theory (2018), Problem 2.84. Original source indexed by AMR-022-2084. https://arxiv.org/abs/1809.07200.