Julia Set: Chaos on Irrational Points on Unit Circle

In summary, the conversation discusses the Julia set, a mathematical set that can be generated by iterating a complex quadratic polynomial. The set is chaotic for non-rational points and has two Fatou domains, the interior and exterior of the unit circle. The conversation also touches on the concept of Fatou domains and how they are open and invariant under the iteration function. An example of an irrational point, (cos(1),sin(1)) or e^{i}, is given to demonstrate chaotic behavior on the Julia set.
  • #1
gursimran
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0
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?

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  • #2
gursimran said:
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: [itex]e^{i}[/itex]. This is a point whose behaviour ought to be chaotic...
 
  • #3
micromass said:
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: [itex]e^{i}[/itex]. 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
 

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1. What is a Julia Set?

A Julia Set is a mathematical set of complex numbers that, when iterated using a specific formula, exhibit chaotic and unpredictable behavior. It is named after the French mathematician Gaston Julia who studied these sets in the early 20th century.

2. What does the term "chaos" mean in relation to Julia Sets?

In mathematics, chaos refers to a complex and unpredictable behavior exhibited by a system that is highly sensitive to initial conditions. In the context of Julia Sets, this means that a small change in the initial parameters can result in a vastly different and seemingly random output.

3. What is the unit circle and how does it relate to Julia Sets?

The unit circle is a circle with a radius of 1, centered at the origin on a Cartesian coordinate system. In Julia Sets, the unit circle is often used as a boundary for the complex numbers that are iterated to create the set. Points outside of the unit circle tend to diverge towards infinity, while points inside the unit circle tend to converge towards a stable value.

4. How are Julia Sets created?

Julia Sets are created by iterating a complex number using a specific formula called the Julia Set equation. This equation is typically written as zn+1 = zn2 + c, where zn is the initial complex number and c is a constant value. The resulting values are plotted on a complex plane, with the points colored based on whether they converge or diverge.

5. What is the significance of Julia Sets in mathematics?

Julia Sets are significant in mathematics because they represent a complex and beautiful example of chaotic behavior in a simple system. They have also been studied extensively in the field of fractal geometry, as they exhibit self-similarity and infinite complexity. Additionally, Julia Sets have practical applications in fields such as computer graphics, cryptography, and dynamical systems.

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