Critically finite maps with hyperbolic postcritical complement

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 $\mathbb P^n$ be complex projective $n$-space, with $n>1$. The source calls a rational self-map $f$ critically finite if there are possibly reducible hypersurfaces $V\subset W\subset\mathbb P^n$ for which \[f:\mathbb P^n\setminus W\longrightarrow\mathbb P^n\setminus V\] is a covering map. A complex manifold is Kobayashi hyperbolic when its Kobayashi pseudodistance separates distinct points.

Statement recorded in the supplied source.

Are there nontrivial examples of such critically finite rational maps for which $\mathbb P^n\setminus V$ is Kobayashi hyperbolic? If so, describe their dynamics. [S2]

Short English statement

Find nontrivial critically finite projective maps with a Kobayashi-hyperbolic postcritical complement, and describe their dynamics.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 5300089 (AMR-052-0089). 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, Critically finite rational maps on $\mathbb P^n$, Problem, printed p. 12. https://arxiv.org/abs/math/9201271v1.