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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 .