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September 17, 2019

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.

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