Given a function f one can work on two different problems:

The orbits of N the associated Newton function to f.

The orbits of f.

Definition: The Julia set of a function F is the set of points that under iteration of F

do not converge to a fixed point

do not converge an attractive orbit.

do not go to infinity

Definition

The set of points whose orbits are bounded under iteration of fc(z)=z^2+c is called the filled Julia set of fc and denoted by Kc. The boundary of the filled Julia set is the (already defined) Julia set.

Remark,

The Julia set of a general function (a function that is not necessarily of the form f(z)=z^2+c) is defined as in two slides ago.

The particular case of a julia set of a function of the form f(z)=z^2+c is defined as the previous slide.

Question:

We iterate the function fc(z)=z^2-0.12+0.75i.

How do we really know if an orbit goes to infinity?

Maybe the first 50 (or 100 or 1000000) iterations get farther from the origin but how do we know the orbit does not “double back” and come back to 0?

The escape value

Proposition: Let fc(z)=z^2+c. If there is a point in the orbit of u with absolute value larger than 2 and larger than the |c| then the orbit of u will go to infinity.

Proposition:

Let f(z)=z^2+c. Either the orbit of a complex number under iteration is bounded or the orbit goes to infinity.

The filled Julia set is the set of points whose orbit does not go to infinity under iteration.

Definition: Let u be a fixed point and let λ=f'(u)

A fixed point is called

attracting if |λ|<1

supper-attracting if λ=0

repelling if |λ|>1

neutral if |λ|=1

Definition: Let u be a periodic point of period k and let λ=(f^k)'(u)

A periodic point is called

attracting if |λ|<1

supper-attracting if λ=0

repelling if |λ|>1

neutral if |λ|=1

Proposition (Fatou, Julia)

If 0 is in the filled Julia set of a function fc(z)=z^2+c (that is, if the orbit of 0 remains bounded under iteration of fc) then the Julia set is connected

On the other hand, if 0 is not in the filled Julia set, then this set is a Cantor set.

Definition:

A set A is connected if given any two points in A there is a path between them inside A).

Definition (hard, can be skipped)

A set of complex numbers C is a Cantor set if

1. C is closed

2. C is bounded

3. C is totally disconnected (this means that for each pair of points in C there no exists path between these two point completely inside C)

4. Every point in C is an accumulation point of C.

Definition

The Mandelbrot set M consists in the set of complex numbers c for which the orbit of 0 of the function

fc(z)=z ^2 +c is bounded (that is, it does not go to infinity)

Why is the orbit of 0 so “important”?

Given a function f one can work on two different problems:

2006-03-29T18:57:45