Chaotic complex dynamics and Newton's method

Question

I'm trying to understand Harold E. Benzinger, Scott A. Burns, and Julian I. Palmore's work on one-parameter families of Julia sets arising from Newton's method in the complex domain. They show the existence of bifurcation points where zeros coalesce or change from attractors to repellors, and points where chaotic behavior occurs.

They write, "Additional information can be obtained by a careful choice of color assignments. If the graphics device is capable of displaying different shades of a particular color, then the shade of color assigned to each starting point can be chosen to indicate the number of iterations required to satisfy. Thus, relative rates of convergence within each basin of attraction can be observed graphically. Backward iteration allows constructing graphically the boundaries between basins of attraction. Rather than performing the usual Newton method...we reverse the process...where the starting point $w_0$ is on the boundary." See:

Full reference:

• Harold E. Benzinger, Scott A. Burns, Julian I. Palmore, Chaotic complex dynamics and Newton's method, Physics Letters A Volume 119, Issue 9, 19 January 1987, Pages 441-446, https://doi.org/10.1016/0375-9601(87)90412-9

Did they just repeatedly iterate $g^{-1}$ on all the points they suspect to be in the basin of attraction in order to find its boundary? It says that the starting point $w_0$ is on the boundary, but isn't the point to find the boundary from the start?

What am I missing here?

Thank you.

Answer 1

"Repeatedly iterating $g^{-1}$" does not really make sense, since your map is not injective.

What is usually done when one talks about "backwards iteration" in this context is that you choose one of the $d=\deg(g)$ preimage points at random, and continue in this way. If you started in the Julia set (the locus of unstable/chaotic behaviour), this will give you a sequence of points in the Julia set. If you start ANYWHERE (well, with at most two exceptions on the sphere), then you get a sequence that will accumulate on the entire Julia set.

But of course you can only draw finitely many points, and a problem with this method is that there will in general be parts of the Julia set that are not well-reached by it. Your points will end up equidistributed with respect to the measure of maximal entropy, and some parts of the Julia set will not receive much of this measure.

The paper you cite is of course rather old, and appeared in a physics journal, and I am not entirely sure that it represented the state of the art even at the time of writing. Hubbard was making pictures of the iteration of Newton's method a long time before this; by all accounts this inspired Mandelbrot to do his computer experiments that led to his popularisation of the "Mandelbrot set". The paper lists no references to contemporary complex dynamics, which was an extremely active field at that point.

In any case, there are now certainly better methods for drawing Julia sets, depending on your specific application. In the case where your function (here: the Newton's method) is hyperbolic, there is even a theorem that says that the Julia set can be approximated to any desired accuracy in polynomial time; see "Hyperbolic Julia sets are poly-time computable" by Mark Braverman.