Maximizing quadratic pressure

Definitions and mathematical statement

Let $p_c(z)=z^2+c$ for $c\in\mathbb C$, and let $\mathcal H$ be the hyperbolic parameter set: the orbit of its critical point $0$ tends to an attracting periodic orbit or escapes to infinity. The Julia set $J_c$ is the complement of the normality set of the iterates on the sphere. For $c\in\mathcal H$, the map is expanding on $J_c$ and has no critical point there.

For a base point $w\in J_c$, define \[ P_c=\lim_{m\to\infty}\frac1m\log \sum_{\substack{z\in J_c\\p_c^m(z)=w}} |(p_c^m)'(z)|^{-1}. \] All $2^m$ preimages are counted; on this expanding set they are distinct. The resulting pressure, for the potential $-\log|p_c'|$, is independent of the chosen Julia-set base point. Put $P_*=\sup_{c\in\mathcal H}P_c$.

Determine which parameters maximize or nearly maximize this pressure. Does a hyperbolic parameter attain $P_*$? If not, describe maximizing sequences $c_j\in\mathcal H$ with $P_{c_j}\to P_*$ and the possible locations of their limiting parameters in the complex parameter plane. Quantitative identification of regions where $P_c>P_*-\varepsilon$, for arbitrary $\varepsilon>0$, is part of the localization question. Both bounded and escaping critical orbits are included. Numerical evidence for a large value at one parameter is insufficient to establish global maximality over all hyperbolic components.

Short English statement

Where in quadratic parameter space is the hyperbolic pressure largest, or arbitrarily close to largest?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3700032 (AMR-036-0032), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Eremenko - My favorite unsolved problems, Littlewood constants, Question D. Original source indexed by AMR-036-0032. https://www.math.purdue.edu/~eremenko/uns1.html.
• S3 A. Eremenko, Some constants coming from the work of Littlewood, Question D. Author-maintained primary problem note. https://www.math.purdue.edu/~eremenko/dvi/lit.pdf.