In today’s post I wanted to quickly highlight a cool relationship between Mandelbrot and Julia sets.
Consider the function, which depends of complex parameter z:
f(z) = x2 + z
Fixing this z, f(z) defines a map from the complex plane to itself. We can start from any value of x and apply this function over and over, which would give us a sequence of numbers. This sequence can either go off to infinity, or not. The boundary of the set of values of x where it doesn’t is the Julia set for this particular z, which we fixed initially.
Conversely, starting with x = 0, we can draw the set of numbers for which the resulting sequence does not go off to infinity. This is called the Mandelbrot set. (Note the subtle difference between the two).
Okay, so the cool relationship is that, near the number z, the Mandelbrot looks like the Julia set for the number z, or as Wikipedia describes:
“There is a close correspondence between the geometry of the Mandelbrot set at a given point and the structure of the corresponding Julia set.”
To illustrate this, consider the following Julia set:
Zooming into the Mandelbrot set at the same value of z gives us this image:
They are extremely similar! So, essentially, the Mandelbrot sets looks like a lot of Julia sets! (Click here to explore this in more detail).
This amazing result is used in lots of results on the Mandelbrot set, for example, it was exploited by Shishikura to prove that “for a dense set of parameters in the boundary of the Mandelbrot set, the Julia set has Hausdorff dimension two, and then transfers this information to the parameter plane“.