Combinatorial theory for geometrically finite 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.Critically finite here means $P(f)$ is finite. Combinatorial equivalence compares $f$ with another map $\widetilde f$ using orientation-preserving sphere homeomorphisms $h_0,h_1$ with $h_0\circ f=\widetilde f\circ h_1$, which identify the closed postcritical sets and are isotopic relative to $\overline{P(f)}$. The sought data are finite topological data.

Statement recorded in the supplied source.

Extend Thurston’s combinatorial theory from critically finite rational maps to all geometrically finite rational maps: classify $f$ up to combinatorial equivalence relative to $\overline{P(f)}$ by a finite amount of topological data, and characterize the combinatorial types realized by rational maps. [S2]

Short English statement

Give finite topological classification data for all geometrically finite rational maps and determine which data rational maps realize.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 5300087 (AMR-052-0087). 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 4, printed p. 12. https://arxiv.org/abs/math/9201271v1.