# Julia Set Distance Estimate

# 2.1 Problem

There is a problem with the third formula when implemented on a computer: “spots” at pre-images of the attracting orbit, worse for strong (but not super) attraction.

See my decade-old math.stackexchange question (accessed 2025-02-20).

For example, the Julia set of \(z \to z^2 + 0.01\):

# 2.2 Derivation Using Koenigs Linearisation

See Chapter 6 of:

Dynamics in one complex variable: introductory lectures, John W. Milnor, 1990, https://arxiv.org/abs/math/9201272

Corollary 6.4 introduces \(\phi(z) = \lim_{n \to \infty} (z_n - \hat z) / \lambda^n\) as linearisation near an attractive fixed point with multiplier \(\lambda\).

When using this for distance estimation, one calculates \(\phi / \phi'\), and the \(\lambda^n\) factor cancels.

However, as Milnor points out,

Note that \(\phi\) has zeros at all iterated preimages of \(\hat z\), and critical points at all iterated preimages of the critical point \(\omega\). The function \(z\mapsto\phi(z)\) is unbounded, and oscillates wildly as \(z\) tends to \(J = \delta\Omega\).

This means that \(\phi / \phi'\) is zero and infinite at various points inside the attractive basin \(\Omega\), and thus it isn’t really useful as a distance estimate.

# 2.3 Doomed Attempt Using L’Hospital’s Rule

L’Hospital’s Rule for limits says that (under some conditions) if \(\lim\frac{f}{g}\) cannot be evaluated as \(\frac{\lim f}{\lim g}\), then it might be evaluated as \(\lim \frac{f'}{g'}\).

Applying L’Hospital’s Rule to the third formula gives a sequence of distance estimates:

\[\begin{aligned} \delta_0(z) &= \lim_{n \to \infty} \frac{|z_{pn} - z_*|}{\left|\frac{\partial}{\partial z} z_{pn} \right|} \\ \delta_1(z) &= \lim_{n \to \infty} \frac{\left|\frac{\partial}{\partial z} z_{pn} \right|}{\left|\frac{\partial^2}{\partial z^2} z_{pn} \right|} \\ \delta_2(z) &= \lim_{n \to \infty} \frac{\left|\frac{\partial^2}{\partial z^2} z_{pn} \right|}{\left|\frac{\partial^3}{\partial z^3} z_{pn} \right|} \\ &\vdots \end{aligned}\]

Then define a combined distance estimate as the maximum of all of these

\[\delta(z) = \max\{ \delta_i(z) : i = 0, 1, 2, \ldots \}\]

In practice, only a finite number can be calculated.

This is unprincipled as the pre-conditions for l’Hospital’s rule are not checked, and it’s not obvious that taking the maximum is always the right thing to do.

Anyway it doesn’t really work: the problem is not the evaluation of the limit, but that the value of the limit is bad as a distance estimate.