Arithmetical Dynamics: Part 0

Arithmetical dynamics is the combination of dynamical systems and number theory in mathematics.

We are here with the Part 0 of the Arithmetical Dynamics Series. Let's get started....

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 .

Make sure you visit the Introduction to Arithmetical Dynamics post of this Series.

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