This question is motivated by the coloring schemes of the Mandelbrot fractal, namely that the color is determined by the points outside the set, and is proportional to the number of iterations $n_c$ it takes for the mapping, $z_{n+1} = z_{n}^2 + c$ to escape the closed disk $|z_{n_c}|>2$ for a fixed $c\in\mathbb{C}, z_0=0$.
It's observed (and perhaps easy to show?), that the closer one gets to the boundary of the set the larger $n_c$ is. Is there an explicit construction for a complex number $c$ that has an arbitrarily large $n_c$? If one exists, is there one that doesn't involve a perturbation off the real axis?
Best Answer
Why perturb at all? It's easier to analyse if you stay on the real axis.
By consideration of the roots of $z^2 - z + c$, the easy construction is $c = \tfrac14 + \varepsilon$. Without the $\varepsilon$ it would converge to $z \to \tfrac12$; $$(\tfrac12 - \delta + \alpha\varepsilon + O(\varepsilon^2))^2 + \tfrac14 + \varepsilon = \tfrac12 - (\delta - \delta^2) + (\alpha + 1 - 2\delta\alpha)\varepsilon + O(\varepsilon^2)$$ so $n_c \approx \tfrac{3}{2\varepsilon}$