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Author Topic:  Unstable Julia sets - still fractals?  (Read 357 times)

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Offline marcm200

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Unstable Julia sets - still fractals?
« on: April 28, 2019, 10:55:26 AM »
The Julia set for c=-0.25+0.75i looks nice at maxit=20, but vanishes at maxit 2000, only exterior. If I zoom in on the maxit=20 version 200fold, I still see sub-structure.

So I was wondering: Will this substructure continue forever - or will it stop at some point? Julia sets are defined as being the mathematical limit of always bounded orbits, but here it looks like no orbit stays bound forever, but the image shows self-similarity and some coast-line structure hinting to a box-dimension of greater than one - at that maxit.

Stopping at a finite maxit leaves one with a polynomial of some (large) degree, but it is fixed. Can that function show fractal-like behavior - or is continous iteration a prerequisite for that?

Linkback: https://fractalforums.org/index.php?topic=2780.0

Offline pauldelbrot

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Re: Unstable Julia sets - still fractals?
« Reply #1 on: April 28, 2019, 12:47:13 PM »
No. The curve eventually becomes smooth and featureless, and the fractal dimension exactly 1 for its boundary.

The true Julia set here is going to be a dust, with a dimension likely below 1. Distance estimator could be used to make it visible.

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