I was recently reminded of the following cute fact which I will state as a proposition to fix notation:

PropositionGiven $\epsilon > 0$, let $c = -3/4 + \epsilon i \in \mathbb{C}$ and $q_c(z) = z^2 + c$. Define the sequence of polynomials $q_c^n$ inductively by $q_c^0(z) = z$ and $$ q_c^{n+1}(z) = q_c(q_c^n(z)). $$ Since $q_c^n(0) \to \infty$ as $n \to \infty$, $$ N(\epsilon) = \min\{n ~:~ |q_c^n(0)| > 2 \} $$ is well-defined. The cute fact is that: $$ \epsilon N(\epsilon) \to \pi, $$ as $\epsilon \to 0$.

The next-simplest example of this phenomenon uses $c = 1/4 + \epsilon$. If we define $N(\epsilon)$ similarly then this time it turns out that: $$ \sqrt{\epsilon} N(\epsilon) \to \pi, $$ as $\epsilon \to 0$.

The points $-3/4$ and $1/4$ are the "neck" and "butt", respectively, of the Mandelbrot set and there are various other examples of these "$\pi$ paths": they are suitably-parameterised curves simultaneously tangent to two bulbs of the Mandelbrot set where they meet it. In fact I'd guess any part of the boundary where a bulb bubbles off works and so there are infinitely many such paths. External rays seem the obvious paths and they have a natural parameterisation. (Note that although $\epsilon \mapsto -3/4 + \epsilon i$ is not an external ray, it is asymptotic to one, as least set-wise, which is really all that matters.)

If true, I'm sure this is all in the literature (perhaps implicitly) but I don't know the field and after of searching through some lovely papers I couldn't find what I wanted. A naive guess is that something like the following might be true:

Half-serious conjectureLet $M \subset \mathbb{C}$ be the Mandelbrot set, $\overline D \subset \mathbb{C}$ the closed unit disc and $\Phi : \mathbb{C} - M \to \mathbb{C} - \overline D$ the unique conformal isomorphism such that $\Phi(c) \sim c$ as $c \to \infty$. Let $e^{i\theta} \in S^1$ (for $\theta$ rational) and let: $$ \alpha : (1, \infty) \to \mathbb{C} - M\\ r \mapsto \Phi^{-1}(re^{i\theta}) $$ Let $c = \alpha(r)$ and define $N(r)$ as above, then: $$ N(r) \sim \pi / f_\theta(r) $$ for some function $f_\theta$, such that $f_\theta(r) \to 0$ as $r \to 1$ and in particular $f_0(r) \sim \sqrt{\alpha(r)-1/4}$ and $f_{1/3} \sim -i(\alpha(r)+3/4)$.

Here then, at last, is my question:

QuestionIs some suitably-modified form of the above conjecture correct and is there a proof in the literature toward which somebody could direct me?

Aside from general curiosity my **motivation** stems from the fact that it's not too hard to prove the stated results for the $c = -3/4 + \epsilon i$ and $c = 1/4 + \epsilon$ paths separately but I'd like to have a unified proof. E.g., to prove the result for the (easier) $c = 1/4 + \epsilon$ path we consider:
$$
z_{n+1} = z_n^2 + 1/4 + \epsilon\\
\Rightarrow z_{n+1} - z_n = (z_n - 1/2)^2 + \epsilon
$$
Then approximate:
$$
\frac{dz}{dn} \simeq (z-1/2)^2 + \epsilon\\
\Rightarrow z(n) = 1/2 + \sqrt{\epsilon}\tan(\sqrt{\epsilon}n)\\
\Rightarrow \sqrt{\epsilon}N(\epsilon) \simeq \tan^{-1}(\frac{3}{2\sqrt{\epsilon}} ) - \tan^{-1}(-\frac{1}{2\sqrt{\epsilon}})
$$
And so provided we can justify the approximation is accurate as $\epsilon \to 0$ (which is tricky but possible) the result follows.