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The Mandelbrot set is formally defined as the values of a complex constantcfor which the Julia Sets of the iterative function:

[tex]

f(z)=z^2+c

[/tex]

areconnected(i.e. consists of a single figure)

Recall that the Julia sets are the launch points [tex]z_0[/tex] of the iteration that gives rise to bound orbits in the complex plan for some given constantc

However, according to some less formal sources (e.g. wikipedia) the Mandelbrot set can also be constructed simply by analyzing the same iterative function, only this time keep the launch point constant (0) and vary the constant - or parameter -cinstead. Those values that give bound orbits are the Mandelbrot Set

This seems like a peculiar accident, but that cannot be so. So my question is: Why does a bound orbit for the iterative function

[tex]

f(z)=z^2+c

[/tex]

launched from [tex]z_0=0[/tex] for a given value ofc, imply that the corrosponding Julia Set is connected?

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# The Mandelbrot Set

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