Julia Sets and the Mandelbrot Set

Newton's Method Basins of Attraction

Basins of z2 - 1

Cayley used an example to introduce the problem of finding the basins of attraction of the roots of a function.
The complex function f(z) = z2 - 1 has two roots,
z = +1 and z = -1.
Denoting the basins of attraction of these roots by A(+1) and A(-1), that is
  A(+1) is all z0 from which Newton's method converges to +1
and
  A(-1) is all z0 from which Newton's method converges to -1
Cayley proved
  A(-1) is the left half of the complex plane, and
  A(+1) is the right half of the complex plane.
In the picture, all points painted black converge to +1, all points painted white converge to -1.
We might expect that the points on the red vertical line are equally attracted to both +1 and -1, and consequently they will not converge at all.
This is true, and on this line iterating Newton's method is a chaotic process.
That is, on the boundary of the basins of attraction, the dynamics are chaotic.

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