Unstable Julia sets - still fractals?

  • 1 Replies
  • 299 Views

0 Members and 1 Guest are viewing this topic.

Offline marcm200

  • *
  • 3c
  • ***
  • Posts: 855
« 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/fractal-mathematics-and-new-theories/28/unstable-julia-sets-still-fractals/2780/

Offline pauldelbrot

  • *
  • 3f
  • ******
  • Posts: 2295
« 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.


xx
Unstable

Started by kohlenstoff on Fractal Image Gallery

1 Replies
175 Views
Last post June 20, 2018, 07:49:15 PM
by Dinkydau
xx
Mandelbulber 2.21 a bit unstable.

Started by kohlenstoff on Mandelbulber

14 Replies
401 Views
Last post August 03, 2020, 09:10:32 PM
by buddhi
clip
It's The Most Unstable, Outdated, Never-Made-It-Out-Of-Beta Program You Need!

Started by Sockratease on Mutatorkammer

2 Replies
576 Views
Last post February 27, 2018, 10:27:43 PM
by Sockratease
clip
Parabolic Julia sets

Started by marcm200 on Fractal Mathematics And New Theories

82 Replies
3400 Views
Last post November 11, 2020, 02:29:07 PM
by marcm200
xx
Embedded Julia sets

Started by pauldelbrot on Image Threads

108 Replies
6878 Views
Last post November 16, 2020, 02:26:14 AM
by pauldelbrot