Fractal Related Discussion > Fractal Mathematics And New Theories

 using Misiurewicz points for perturbation and/or series approximation

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claude:
My perturbation experiments have mostly been using the periodic nucleus of a central minibrot as primary reference for perturbation with series approximation.  As is known, glitches occur around minibrots that are not the main reference, they can be detected and corrected.

So I wonder if maybe better results (fewer glitches needing correction, so fewer references total, so faster rendering) would be achieved by using Misiurewicz points as references.  Has anyone tried this already?

Misiurewicz points are in the Mandelbrot set, and are strictly pre-periodic.  They are commonly tips, branch points, centers of spirals.  Simple examples are -2 and i, iterating them by hand should enlighten you what "strictly pre-periodic" means.  They are repelling, while minibrots are attracting.

Misiurewicz points can be found by Newton's method, see my blog post https://mathr.co.uk/blog/2015-01-26_newtons_method_for_misiurewicz_points.html

Another blog post from the same time describes Misiurewicz domains, an analogy to atom domains for periodic points https://mathr.co.uk/blog/2015-02-01_misiurewicz_domains.html

I've been investigating how the periods of the Misiurewicz points behaves in embedded Julia sets, see attached image.  This is a doubly-embedded Julia set, near elephant valley of a period 59 minibrot, near sea horse valley of period 3 minibrot.  The different colours represent different preperiods.  You can see the shape stacking in effect, the period 3 Misiurewicz points are in the seahorse valley spirals, and the period 59 Misiurewicz points are in the elephant valley curves.  Moreover the pre-periods also stack up in a regular way, being also combinations of the "cardioid partials" (here the partials are 1, 3, 59, with a period 537 central minibrot).  The numeric patterns are very interesting, I hope to research them some more.


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gerrit:
Have you tried using Misiurewicz point for the reference orbit?

claude:
no, I haven't investigated this any further yet due to lack of time

gerrit:
Is there a way to avoid exhaustive search of all periods and preperiods, some analogue of the box/triangle method for detecting nuclei periods?

claude:
I haven't tried this yet, so it may fail or be complete nonsense..

What about iterating the box ABCD, and its center point Z, and seeing which iterations of the box surround which other iterations of Z?

If Z is near a periodic point, box ABCD_period surrounds Z_0 = 0+0i.

Perhaps, if Z is near a preperiodic point, box ABCD_{preperiod + period} surrounds Z_{preperiod} ?  period > 0 of course.  For typical embedded Julia sets I think the spirals and tips have lower total (period+preperiod) than the (period) of the low period islands.

Even if this works, it will probably fail for typical embedded Julia set views due to the symmetry - maybe a subdivided grid could be used to isolate the points.  Highly morphed Julia sets might need a fine grid, though...

update it seems to work!  the attached image is an embedded Julia set in the seahorse valley of a period 5 island in the antennae of the 1/3 bulb. I drew small boxes around the spiral centers, and refined with Newton's method.  The probe boxes must be smaller than the boxes for periodic components, else non-structural Misiurewicz points are detected (I don't know if this would matter for references for rendering purposes)
https://code.mathr.co.uk/mandelbrot-numerics/blob/HEAD:/c/lib/m_d_box_misiurewicz.c reference implementation in native precision, an arbitrary precision version may be better storing the central orbit rather than recomputing it.

update 2 seems in embedded Julia sets there is a simple relationship with the periods of the influencing islands, added more pics...

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