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While studying a visual representation the Mandelbrot set, I have come across a very interesting property:

For any point inside the same primary bulb (a circular-like 'decoration' attached to the main body of the set), the periodicity of that point (i.e. the pattern of values that emerges when '$f(x) = z^2 + c$' is iterated with the '$c$' value that represents that point) is constant.

Does anyone know how to prove this property in a mathematical way? Is there more than one way in which this could be shown?

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    I've edited the title to make it more descriptive.2012-06-27
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    Similarly, each bulb contains a root of $f^n(z)=0$. I seem to recall that the number of spokes was a factor of $n$, but it has been a while.2012-06-28
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    @RossMillikan : Do you know how to prove any of these properties?2012-06-28
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    @Brian: no, they were observations of mine long ago. The border of the bulb is where the derivative of the iterative loop is 1 in absolute value. The largest bulb off a root has twice the period of the main bulb, and the next two have three times the period, etc. Essentially the root of a bulb acts like the main bulb. I believe you can prove all this because the function is analytic, but don't have the details.2012-06-28

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