Extremality of iterated quadratic polynomials

Definitions and mathematical statement

For a polynomial $p$, define $I(p)=\int_{|z|<1}|p'(z)|/(1+|p(z)|^2)\,dA(z)$, where $dA$ is Euclidean area. Let \[ \Phi(N)=\sup_{\deg p=N}I(p),\qquad \alpha=\limsup_{N\to\infty}\frac{\log\Phi(N)}{\log N}. \] This exponent measures the largest possible growth, in degree, of the integrated spherical derivative.

For $c\in\mathbb C$ put $p_c(z)=z^2+c$. Call $c$ hyperbolic when the orbit of the critical point $0$ tends to an attracting periodic orbit or to infinity. Write $p_c^m$ for composition iteration, of degree $2^m$, and set \[ a(c)=\limsup_{m\to\infty} \frac{\log I(p_c^m)}{m\log2}. \] The source asks whether iterated hyperbolic quadratics are extremal or nearly extremal for $\alpha$. At the level of growth exponents, a precise form of this question is $\alpha=\sup_{c\ \mathrm{hyperbolic}}a(c)$.

Does this equality hold? Equivalently, can every $\varepsilon>0$ be matched by one fixed hyperbolic parameter $c$ and arbitrarily large iteration numbers whose normalized logarithmic integral exceeds $\alpha-\varepsilon$? If so, determine whether one parameter attains the exponent or only an approaching sequence of parameters does. The integration disk is fixed, and the parameter in $a(c)$ is held fixed while the iteration number tends to infinity. No equality of finite-degree maxima is asserted by this exponent formulation.

Short English statement

Can iterates of hyperbolic quadratic polynomials approach the optimal polynomial growth exponent for integrated spherical derivatives?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3700031 (AMR-036-0031), 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 C. Original source indexed by AMR-036-0031. https://www.math.purdue.edu/~eremenko/uns1.html.
• S3 A. Eremenko, Some constants coming from the work of Littlewood, Question C. Author-maintained primary problem note. https://www.math.purdue.edu/~eremenko/dvi/lit.pdf.