I have been working on a transformation consisting of altering the behaviour of the Mandelbrot algorithm, in such a way that "C" acquires the value of a constant "J" in a given iteration "I". At that moment, the algorithm will stop calculating the Mandelbrot Set and will use the rest of the iterations to calculate the Julia Set, which will be embedded within the Mandelbrot calculated so far and influenced by the value of "Z" after those first iterations.

My tests seems to indicate that the transformation performs well for any function handled by the Mandelbrot method, so it could be a good starting point to complicate it and do cooler things

The most suitable name that I have come up with for them is "Juliter Transformation", and I have tried to explain it to my students in this document: https://fractalfun.es/juliter.php?i=en

I leave you here some of my results.



Linkback: https://fractalforums.org/index.php?topic=4148.0

Had some fun with this transform. I applied it with I = 2500 to the Mandelbrot set itself, see below for cusp of period 3 mini normal and then after the Juliter transform, and then the largest bulb left of the bulb on top. UF code for transform:
JuliterTransform {
; from @sergioct
transform:
  complex func f(const complex w, int it)
     z = 0
     int k = 0
     while (k<it)
       z = z^2 + w
       if(|z|>4)
         z = 1e12
         return z
       endif
       k = k + 1
     endwhile
     return z
  endfunc
  if @isOn
    #pixel = f(#pixel,@it)
  endif
default:
  title = "JuliterTransform"
  bool param ison
    caption = "On"
    default = false
    hint = "Turn on/off"
  endparam
  int param it
    caption = "I"
    default = 16
  endparam
}


On the right you can see the Mandelbrot Set with the Juliter Transformation that corresponds to the Julia Set on the left.

Looks the same (first image). Increasing I from 16 to a large value gives cleaner result (2nd image).

Yes, your second result also matches mine by increasing "I"

I thought I had seen this before.

See https://fractalforums.org/index.php?topic=442.msg2204#msg2204 where this method amongst others is discussed more generally.

Chris M Thomasson (in FB and before that, Google+) has also been doing the Mandelbrot transformation before Julia for some time, and calls it Mulia.

The general principle of pre-transforms seems the same as what is used for water effects, gnarls and other filters. I hadn't thought of doing the Mandelbrot hybrid this way rather than with some integrated method; thanks!

I tried changing the function to newtons iterations (f(z) = z^3 - 1) for iterations >= 8
I also had to change the bailout criterion from escaping to converging (|z - zprev| < 1e-6)

A mandelbrot using juliter on FFExplorer. With juliter on left.

This is a phoenix mandelbrot inverse. The juliter are the most part of the image.