Jordan boundary of a cubic parameter component

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.

For $\lambda\in\mathbb C$, let $f_\lambda(z)=\lambda z^2+z^3$. The point zero is a superattracting fixed point. Its immediate basin is the connected component containing zero of the set of points whose forward iterates tend to zero. Let $U$ be the parameter component containing $\lambda=0$ for which both finite critical points (zeros of $f'_\lambda$) lie in this immediate basin. Take its boundary $\partial U$ in the complex parameter plane. A cusp on $\partial U$ is a parameter for which $f_\lambda$ has a parabolic periodic cycle, whose return-map multiplier is a root of unity. A Jordan curve is the image of a circle under a homeomorphism onto its image.

Statement recorded in the supplied source.

Is $\partial U$ a Jordan curve? Prove this conjecture. [S2]

Short English statement

Prove that the boundary of the designated cubic parameter component is a Jordan curve.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 5300019 (AMR-052-0019). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Curt McMullen, Rational maps and Teichmüller space, in Ben Bielefeld and Mikhail Lyubich (editors), Problems in holomorphic dynamics (1992). Density of cusps; Example III and second Conjecture, p. 10. https://arxiv.org/pdf/math/9205209.