My bad for the delayed reply.
So I'm not exactly sure why they insist on definitions of "prisoner" and "escape" sets here. They are indeed terms that frequently appear in fractal geometry, but they're not conventional terms that everyone's supposed to know how to use.
Loosely speaking,
- Escape occurs if \( \lim_{n\to\infty} |z_n| =\infty \). That is, if you keep iterating the function \(f(z) =z^2+c\) for a given \(c\), over and over (starting with \(z_0 = 0\) in this context), the modulus of the numbers you get will always grow bigger and bigger. The escape set is the set of all complex numbers \(c\) that have this property.
- The prisoner set refers to the set of all complex numbers \(c\) that fail this property. This essentially occurs when if you keep iterating with the function \(f(z)=z^2+c\), the modulus of the numbers does not grow bigger and bigger.
As a consequence, if you find that the modulus of the values of \(z_0\), \(z_1\), \(z_2\), \(z_3\) start growing closer and closer to 0, or start oscillating somehow, they should be in the prisoner set.
(Note that for your example, where \(c=i\), the values themselves literally start oscillating. So you'd expect the modulus to do something similar as well.)