Haken-type decomposition for rational maps

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 a rational map $f:\widehat{\mathbb C}\to\widehat{\mathbb C}$ of degree at least two, $J(f)$ is the complement of its Fatou set, where the iterates form a normal family. If $C(f)$ is its set of critical points, set $P(f)=\bigcup_{n\geq1}f^n(C(f))$. In the source convention $f$ is geometrically finite when $\overline{P(f)}\cap J(f)$ is finite; parabolic cycles are allowed.

Statement recorded in the supplied source.

Develop an analogue, for geometrically finite rational maps, of the Haken decomposition of three-manifolds. In particular, when $J(f)$ is disconnected, can $f$ be constructed by surgery from rational maps with connected Julia sets? [S2]

Short English statement

Develop a Haken-type decomposition for these rational maps, including construction of maps with disconnected Julia sets by surgery from maps with connected Julia sets.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 5300086 (AMR-052-0086). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Ben Bielefeld (editor) and conference contributors, Conformal Dynamics Problem List, arXiv:math/9201271v1 (1990), Curt McMullen, §5, Geometrically finite rational maps, Problem 3, printed p. 11. https://arxiv.org/abs/math/9201271v1.