Expanding conformal metric for nonrecurrent quadratics

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.Let $P_c(z)=z^2+c$ have connected Julia set. Nonrecurrence means that its finite critical point $0$ does not belong to its own $\omega$-limit set, the set of accumulation points of its forward orbit. A conformal metric has length element $\rho(z)|dz|$; integrable singularities are allowed provided the improper local length integrals remain finite. Expansion means uniform increase of lengths by a factor greater than one under an iterate, on the Julia set.

Statement recorded in the supplied source.

Does nonrecurrence of $P_c$ imply the existence of a conformal metric with integrable singularities in which $P_c$ is expanding on $J(P_c)$? [S2]

Short English statement

Does every nonrecurrent quadratic with connected Julia set admit an expanding conformal metric with integrable singularities?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 5300073 (AMR-052-0073). 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), §1, Question 13, printed pp. 1--2. https://arxiv.org/abs/math/9201271v1.