Julia Set: Chaos on Irrational Points on Unit Circle

[gursimran]

This text patch is taken from wikipedia article http://en.wikipedia.org/wiki/Julia_set

"For f(z) = z2 the Julia set is the unit circle and on this the iteration is given by doubling of angles (an operation that is chaotic on the non-rational points). There are two Fatou domains: the interior and the exterior of the circle, with iteration towards 0 and ∞, respectively"

1. I have drawn the julia set and plotted the points to see how they behave on the julia set. But for angles 45 90 they all converge to 0, so f'(z) =0. Isn't this a foutau domain?

2. For points such as pi/3 point oscillate b/w two points 120 and 240 deg.

2. It is written that the behaviour is chaotic for irrational points. can anyone give example of such irrational points on unit circle so that i can look how an chaiotic behaviour is?

Pic attached

 

[micromass]

This text patch is taken from wikipedia article http://en.wikipedia.org/wiki/Julia_set

"For f(z) = z2 the Julia set is the unit circle and on this the iteration is given by doubling of angles (an operation that is chaotic on the non-rational points). There are two Fatou domains: the interior and the exterior of the circle, with iteration towards 0 and ∞, respectively"

1. I have drawn the julia set and plotted the points to see how they behave on the julia set. But for angles 45 90 they all converge to 0, so f'(z) =0. Isn't this a foutau domain?

Can you describe exactly what you think that the Fatou domain is? Remember that the Fatou domain must be open and invariant under f!

2. It is written that the behaviour is chaotic for irrational points. can anyone give example of such irrational points on unit circle so that i can look how an chaiotic behaviour is?

Take (cos(1),sin(1)) for example. Or in complex notation: e^{i}. This is a point whose behaviour ought to be chaotic...

 

[gursimran]

Can you describe exactly what you think that the Fatou domain is? Remember that the Fatou domain must be open and invariant under f!

Fatou domain is everything what is not included in julia set and julia set is one in which all numbers dance around and does not converge anywhere.. that's what i think

Ya I've not totally understood the meaning of open and invariant set. Here what I have understood.

Open set in this context means that the set Fi has infinite elements that are different n following a regular pattern inc or dec in the long run...
Invariant set I'm not very clear .. but it has to be something that f does not operate on fatou domains.. but why?



Take (cos(1),sin(1)) for example. Or in complex notation: e^{i}. This is a point whose behaviour ought to be chaotic...

Ya thanks a lot . Now I'm not getting any regular behaviour. I have attached the pic