Size of the instability locus

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 $\mathcal A$ be an analytic parameter family of rational maps of fixed degree at least two. A parameter is $J$-stable when on a neighbourhood its Julia sets move by a holomorphic motion that conjugates the dynamics: each point varies holomorphically with the parameter and the motion is injective on each Julia set. Let $Q\subset\mathcal A$ be the complementary $J$-unstable locus. Lebesgue measure and Hausdorff dimension are taken in real local coordinates on the parameter space; $\dim\mathcal A$ in the comparison means its real dimension.

Statement recorded in the supplied source.

What is the Lebesgue measure of $Q$? Is $\dim_HQ=\dim_{\mathbb R}\mathcal A$? [S2]

Short English statement

How much ordinary volume does the unstable locus occupy in an analytic rational-map family, and does it have the full Hausdorff dimension of that family?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 5300046 (AMR-052-0046). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Mikhail Lyubich, "Measure and Dimension of Julia Sets", in Ben Bielefeld and Mikhail Lyubich (eds.), Problems in Holomorphic Dynamics (1992), Problem 6, pp. 27; arXiv:math/9205209v1. https://arxiv.org/pdf/math/9205209.