Perturbation at deep zoom levels

[kjknohw]

Perturbation produces interesting effects (such as lobes cut-off https://element90.files.wordpress.com/2012/07/perturbed-3-s20120707-02-1200x800.jpg), but these features are lost at deep zooms. They can never be retrieved by zooming further, even if the formula is modified to include baby m-sets.

However, cos(1/z^2)*exp(1/z)^2*(z-1)^2 fixes this. It produces long-thin mini m-sets that always have a zone of perturbation, even at deep zoom. The transition zone in the middle of this image has the effects of perturbation we know and love, despite this image being a deep zoom.

What is going on with this formula? The (z-1)^2 term simply lets points escape and adds z^2 mini m-sets. It doesn't have a decisive effect otherwise.

This cos and exp terms both create an essential singularity at z=0 (and are 1 at large |z|). The exp term produces large mini-msets but the cos term fights against that, opening things up. There is flexibility in the exact powers of terms we use and the formula seems to be a good compromise.

Much is to be explored with essential singularities, they offer a very complex structure.

[Kalles Fraktaler]

I assume this is not related to the perturbation term we use for describing how we render a fractal from a stored reference, in order to use high precision only for the reference?

[kjknohw]

Correct, this is the perturbation that distorts the Mandelbrot set. Not the amazing find of whoever came up with that method.