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?