It is a root point between period 1 and period 3 hyperbolic components of Mandelbrot set. It can be computed using :
internal angle ( rotational number) = 1/3
internal ray = 1.0
Image shows a zoom into dynamical z-plane centered at the alfa fixed point :
Colors of points :
black = interior of Julia set
green = exterior of Julia set
white = forward orbit of some points of interior ( near fixed point alfa).
White cross shows fixed point alfa
One can see here that :
some points of interior first escapes from alfa fixed point and after that fall into it
exterior ( green points) is very thin ( width smaller then width of the pixel ) near alfa fixed point ( and its preimages)
How to do it ?
Run program mandel by Wolf Jung [2]You are now on parameter plane ( left image) and use complex quadratic polynomial ( map) where c=0.0 ( default setting )
Change c parameter : go to bifurcate point from period 1. Use main menu/Points/Bifurcate or key C to open input window. Enter a quotient ( = internal angle = rotational number ) = 1/3 and press enter. Now c = -0.125000000000000 +0.649519052838329 i. You can see it above parameter window. Period =10000 means here that program have not found the period because of numerical problems. Point c is a root point between period 1 and 3.
Go to the dynamic z-plane ( right image ) : use main menu/File/To dynamics or F2 key. You are now ( yellow cross ) at the critical point : z = 0.000000000000000 +0.000000000000000 i
Go to the alfa fixed point. Use main menu/Points/Find point or x key. Enter number 1 ( period=1 == fixed point ) and press enter. Now you are at point : z = -0.250000000000000 +0.433012701892219 i
Zoom in using z key few times .
increase iterations using main menu/Draw/Iterations ( max = 65 000 )
choose few points near fixed point and draw its orbots using keys Ctrl-F ( press and do not release , because it is a slow dynamic !!! )
to share – to copy, distribute and transmit the work
to remix – to adapt the work
Under the following conditions:
attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.