I previously wrote about
Mandelbrot set Newton basins
in the context of finding islands, whose nuclei are periodic points. A
periodic point of a function **g** satisfies
**g ^{p} = g^{0}**, where

**p**is called the period. For the Mandelbrot set the function is

**f**. This post is about preperiodic points that satisfy

_{c}: z → z² + c**g**, where

^{p+k}= g^{k}**p**is the period (still) and

**k**is the called the preperiod.

Newton's method tries to find a root of a function **h(x) = 0**
by iterating **x → x - h(x) / h'(x)** where **h' = dh/dx**,
namely the differential of **h** with respect to **x**.
Here the quadratic function **f** in the Mandelbrot set is
considered as a function of **c**, with **f' = df/dc**
and we want to find a preperiodic **c _{0}** satisfying

**f**.

_{c0}^{p+k}= f_{c0}^{k}Now, **f** and **f'** can be computed by
recurrence relations:

F_{c}^{0}= 0

F'_{c}^{0}= 0

F_{c}^{n+1}= (F_{c}^{n})² + c

F'_{c}^{n+1}= 2 F_{c}^{n}F'_{c}^{n}+ 1

Applying Newton's method gives:

c → c - (F_{c}^{p+k}- F_{c}^{k}) / (F'_{c}^{p+k}- F'_{c}^{k})

But solving this isn't the whole story - it might converge to a preperiodic
point with a preperiod less than **k**, say **k'**.
(Even the target period **p** may be a multiple of the true period,
say **p'**.) The next step is to find the true preperiod of the
resulting **c _{0}**, which can be done by finding the
smallest

**k**such that the Newton's method iteration starting with

**c**still converges to

_{0}**c**itself and not some other point.

_{0}Enough of how (for full details read the source linked below), here are some images, each with a certain fixed period and coloured according to the true preperiod of the root converged to at each pixel.

Image *a* shows the whole Mandelbrot set with some preperiodic
basins of period 1 highlighted. You can see they surround some terminal
and branch points, but not all. Images *b* and *c* show
enlarged regions near the 1/3 and 2/5 bulbs. Image *d* starts to
get interesting - this is zoomed in near the 1/3 child of the 1/2 bulb.
Notice how only the outer filaments have basins attached. Compare with
image *e* which increases the target period to 2: here the inner
filaments have basins attached.

This leads me to conjecture that multiplicative *tuning* is at
work: the inner filaments near a child atom will have preperiodic branch
points that have a period a multiple of the parent atom's period, compared
to the corresponding preperiodic branch points at the root. This seems
to be supported by the remaining images: *f*, *g*, *h*
with periods 1, 2, 3 highlighted near the period 3 island; *i*
near a period 4 island with period 4 highlighted, and *j* near a
period 5 island with period 5 highlighted. Note the inner filaments
being highlighted when the periods match.