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.
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.
