Derivative at infinity on the Riemann sphere

[Unit]

I am reading a recent (2003) paper, "Fatou and Julia Sets of Quadratic Polynomials" by Jerelyn T. Watanabe. A superattracting fixed point is a fixed point where the derivative is zero. The polynomial P(z) = z² has fixed points P(0) = 0 and P(∞) = ∞ (note we are working in \hat{\mathbb{C}} = \mathbb{C}\cup\infty). The derivative at 0 is P'(0) = 0. The author then writes "the derivative of P at ∞ is given by 1/P'(1/ζ) = ζ/2. Evaluating at ζ = 0 gives P'(∞) = 0."

What's going on here? We seem to have conjugated P' by the Mobius map \zeta\mapsto1/\zeta. Why? I would think, if anything, we'd have to conjugate P itself, in order to find properties of P...

 

[Unit]

Oh, I think I've gotten somewhere.

Suppose P(a) = a. Let φ be a Mobius map. With b = φ(a), the derivative of φPφ^-1 at b is φ'(P(φ^-1(b))) P'(φ^-1(b)) / φ'(φ^-1(b)) = φ'(P(a)) P'(a) / φ'(a) = φ'(a) P'(a) / φ'(a) = P'(a).

So we can get P' at a fixed point a by evaluating the derivative of φPφ^-1 at b = φ(a). Let P(z) = z² + c with fixed point ∞, and φ(z) = 1/z = φ^-1(z) so that φ(∞) = 0. Then φPφ(z) = φP(1/z) = φ(1/z² + c) = z²/(cz²+1), whose derivative is 2z/(cz²+1)², which evaluates to 0 at z =0. So P'(∞) = 0, apparently.

I don't quite understand this. And this still doesn't explain the conjugation of P' by 1/z in the author's example. Can somebody shed some light?

 

[Hurkyl]

Aside: you have the fixed points of P wrong, unless you were assuming c=0.

 

[Unit]

Hurkyl: oops, thanks! I'll fix that.

 

[Unit]

It appears to be defined thus. See Beardon's Iteration of Rational Functions, pp. 40-41.

Also, http://en.wikipedia.org/wiki/Periodic_points_of_complex_quadratic_mappings.

Thanks for your time!