Newton's Method and Julia sets and the Douady rabbit. What's the relation?

Question

I have question about Newton's method and the Douady rabbit.

I know that :

• The Douady rabbit is a Julia set with a specific choice of the complex number $c$ in the polynomial $z^2+c$

• The rabbit is the set of all initial points for which Newton's method does not converge to any root in the polynomial $P(x) = (z-a+1/2)(z+a+1/2)(z-1)$.

But I don't understand what's the relation between them ?

And what is the relation between the two polynomials ?

Also, what's the relation between the convergence of Newton's method and the convergence of the norm of $z$ in the Julia set?

Answer 1

what's the relation between the convergence of Newton's method and the convergence of the norm of zz in the Julia set?

Douady rabbit is a set of nonescaping points which is the basin of attraction of period 3 orbit:

$$\{z: f_c^3(z)=z\}$$

One can find such orbit using :

• Newton ( iterative) method : $z_{n+1}=z_n-{f_c(z)\over f_c'(z)}$
• iterating : $z_{n+1} = f_c(z_n)$ and checking its norm ( bailout test ) because if $|z|>2$ then z is allways escaping ( it is called escape time method )

Zeros of both methods should be the same but basins will be different because these are different iterative methods of finding roots