The shifty conjugacy

test-prep

#1

Suppose that f has a fixed point at z_0. Show that the function g obtained by conjugating f with the function \varphi(z)=z+z_0 has a fixed point at zero. In addition, show that the conjugation preserves the nature of the fixed point as attractive, super-attractive, repulsive, or neutral.


#2

Suppose f(z_0) = z_0.

Let \varphi(z)=z+z_0.

Then f(\varphi(z)) = f(z+z_0).

So f(\varphi(0)) = f(0+z_0) = f(z_0) = z_0.

So there is a fixed point at zero.

Now we will show that conjugation preserves the nature of the fixed point.

First note that (f(\varphi(z)))' = \varphi'(z)\cdot f'(\varphi(z)) = f'(z+z_0).

So then |(f(\varphi(0)))'| = |f'(z_0)|.

So the nature of the fixed point will be preserved.