Rational function \( R(z)= \frac {P(z)}{Q(z)} \) ; where P and Q are polynimials . There are some theory about fixed points .

Theorem:

Let \( \rho \) be the fixed point of the maps R and g be the Mobius map . Then \( gRg^{-1} \) has the same number of fixed points at \( g(\rho) \) as \( R \) has at \( \rho \).

Theorem :

If \( d \geq 1 \) , a rational map of degree d has previously \( d+1 \) fixed points in .

To each fixed point \( \rho \) of a rational maps R , we associate a complex number which we call the multiplier \( m(R , \rho) \) of R at \( \rho \) .

$$ m(r, \rho) = \{ R^{,}(\rho) ; \ if \ \rho \neq \propto \ and \ \frac {1}{R^{,}(\rho)} ; \ if \ \rho = \propto $$

Now, we dive into classification of fixed points and this is purely local matter , it applies to any analytic function and in particular , to the local inverse(when it exists ) of a rational map .