What is a prisoner set + Escapee Set In Fractal Geometry

Question

Are both prisoner sets and escapee sets only applicable in Mandelbrot sets, and does prisoner sets have to converge to zero or to any real number and do escapee sets have to converge to infinity?

Thank you so much!

Answer 1

I have not heard those terms used, but I can deduce their meaning. In general for escape time fractals (which include the Mandelbrot set, the "Burning Ship", "Nova" fractal, and other variations - basically anything where you iterate a formula for each pixel independently and colour it according to some outcome) there are a few ways points can behave:

1. converge to a periodic cycle

   1. the period can be $1$, i.e. a fixed point
   2. one of the points in the limit cycle can be $\infty$, i.e. typically called "escape", if it is a fixed point

2. don't converge to a cycle with finite period

   1. chaotic behaviour / strange attractor
   2. Siegel disc as one example (parabolic cycle with irrational rotation number)

For some formulas (rational functions for example), $\infty$ can return back to small values, so the concept of "escape" being "once big its big forever" is not universal.