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# Arithmetical Dynamics: Part 1

We are here with the Part 1 of the Arithmetical Dynamics Series. Let's get started....

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

#### Definition:

Suppose that $\zeta \in C$ is a fixed point of an analytic function $f$ .

Then $\zeta$ is :

a) Super attracting if $f^{'} (\zeta) =0 \rightarrow$ critical point of $f$

b) Attractting if $0 < |f^{'}( \zeta )|< 1 \ \rightarrow$ not a critical point of $f$

c) Repelling if $|f^{'}( \zeta )|>1$

d)Rationally indifferent if $f^{'}( \zeta )$ is a root of unity .

e) Irratinally indifferent if $|f^{' }( \zeta)|=1$ , but $f^{'}( \zeta )$ is not a root of unity .

R has a period n ; $R ^ {n} ( \zeta ) = \zeta$ .

If we denote $R^m(\zeta) = \zeta-m ; m= 0, 1 ,2 ,3 .......$ .

So $$\zeta_{m+n}= \zeta_m \ then \ \ (R^n)^{'}( \zeta )= \prod_{i=0}^{n-1} ( \zeta_k ) \ [fixed]$$

So , we can say about attractuing , sup-attracting , repelling of $R^n$ in terms of multiplier of $R^n$ .

(Super)attracting points (cycles) relate to Faton set .

(Repelling) points (cycles) relate to pulin set .

Make sure you visit the Arithmetical Dynamics Introduction post of this Series before the Arithmetical Dynamics Part 1.