We end up with a function g of w equals w squared,

which we already understand under iteration.

So here's our situation.

F of z, z squared- 2, phi is this function w + 1 over w, and

we just calculated that phi inverse of f of phi of w is just w squared.

Or, if you wanted to solve that for f, we bring the two phis onto the other side of

the equation so, bring the phi inverse here and bring that over here.

Then you get that phi of g of phi inverse of z is f(z) and

g is that function of z squared.

So on the outside of the interval from minus to the 2,

the function f that we want to understand, behaves exactly

like the function g behaves on the outside of the closed unit disk.

But, on the outside of the closed unit disk, the iterates

of the function g just go to infinity.

That's part of the Fatou set.

And so, therefore we can conclude that the outside of the interval from -2 to 2

is part of the Fatou set for the function f and

is actually the set that gets attracted to infinity under f and

therefore A of infinity the base of attraction of infinity for

the function f is everything but the closed integral from -2 to 2.

Since we know that the boundary of this set gives us the Julia set,

we know the Julia set is the closed interval from -2 to 2.

We have looked at two examples so far, and found their Julia sets.

We looked at f(z) = z squared, and

found the Julia set is the unit circle, and we looked at z squared- 2,

and we found the Julia set is the closed interval from -2 to 2 on the real axis.