Why more iterations benefit deeper Mandelbrot zooms over shallow zooms?

Question

When rendering the Mandelbrot set fractal, we set a maximum number of iterations to test each point.

If the escape criteria are met within the maximum iterations, we can stop further iterations because we are sure the point diverges. If the escape criteria are not met, we can guess point does not diverge.

In practice, it is observed that deeper zooms (smaller viewport) require a higher maximum number of iterations to mitigate against blurring of the detail.

The blurring occurs because the error is asymmetric. Too few iterations means points are misclassified as not diverging, they are not misclassified as diverging.

Question: Why does an increase in the number of iterations benefit deeper zooms more than shallow zooms?

Thoughts:

My inexpert brain tells me the key factor is the distance between two points. In deep zooms, the distance is small. In shallow zooms, the distance is larger.

Surely points can be misclassified at the same rate/probability for shallow zooms as much as for deep zooms?

In addition, since the dynamics are chaotic and sensitive to initial conditions, surely we can't say two very very close points are more or less likely to be in different domains (diverge, not diverge) than distant ones?

Answer 1

As you seem to know from this question, once the $n^{\text{th}}$ iterate of $f_c$ from the critical point zero exceeds two in absolute value, we know that each subsequent iterate can only get bigger. Symbolically, $$|f_c^n(0)| < |f_c^{n+1}(0)|,$$ whenever $|f_c^n(0)|>2$. As a result, $$\{c\in\mathbb C:|f_c^{n+1}(0)| \leq 2\} \subset \{c\in\mathbb C:|f_c^{n}(0)| \leq 2\}.$$ In words, if the critical orbit for a particular value of $c$ has not escaped after $n+1$ iterates, then it certainly couldn't have escaped after $n$ iterates.

Let's now write $$E_n = \{c\in\mathbb C:|f_c^{n}(0)| \leq 2\}.$$ The set containment above could then be written $E_{n+1} \subset E_n$. Now, the Mandelbrot set itself could be defined as $$\bigcap_{n=1}^{\infty} E_n.$$ That is, the Mandelbrot set consists of all those $c$ values such that the $n^{\text{th}}$ iterate hasn't escaped for any $n$. Furthermore, the set containment indicates that these sets collapse down to the Mandelbrot set. We can even plot these sets $E_n$ together:

All this leads to the conclusion that:

The closer a point is to the Mandelbrot set, the more iterates you generally need to detect if it is in the exterior or not.