Problem : How changes distance between 2 near points under iteration ? Can I tell to which set these points belong when I know it ?

measure the distance on the complex plane

Here are 3 diagrams . Each show distance between 2 points on the complex plane of :
* exterior of Julia set ( basin of attraction of infinity)
* Julia set
* interior of Julia set

distance between 2 points z2 and z1, measured on complex plane = abs(z2-z1)

c = 0
Initial distance between point is : 0.0062831749717591
Radius is :
1.2 for exterior
1.0 for Julia set
0.8 for interior.

( Julia set for c=0 is a unit circle, where radius =1 and center z=0 )

Angles are : 0.003 and 0.004 (These are 3 pairs of points)

Point that are close in the begining after n iterations :
* are teared apart = distance between them goes to infinity ( these points belong to the exterior)
* distance is changing but is always smaller then distance between beta and -beta (for c=0 it is 2) ( Julia set)
* points become more closer ( distance between them goes to zero after some peak which I do not understand) ( interior points)

measure the distance on the Riemann sphere

"As distance between two points in the complex plane we take their distance on the sphere, i.e. the length of the orthodrome. This means that the metric is bounded by pi and even the distance to infinity (which is now the north pole) is finite." (Georg-Johann)
"the action of the iterated functions on near points is examined. Places, where points, which are near enough, remain near during iterations, belong to the Fatou set. Places, where points, as near they may be, are teared apart, belong to the Julia set." ( Michael Becker )

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Autor: Adam Majewski
Feel free to e-mail me. (:-))