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# Arithmetical Dynamics: Part 3

We are here with the Part 3 of the Arithmetical Dynamics Series. Let's get started....

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

#### Theory:

Let $\{ \zeta_1 , ......., \zeta_m \}$ be a ratinally indifferent cycle for R and let the multiplier of $R^m$ at each point of the cycle be $exp \frac {2 \pi i r}{q}$ where $(r,q) =1$ . Then $\exists \ k \in Z$ and$mkq$ distinct component $F_1 , F_2 , ..... , F_{mkq}$ s.t. at each $\zeta_j$ there are exactly $kq$ of these component containing a petal of angle $\frac {2 \pi}{kq} \ at \ \zeta$ .

Further R acts as a permutation J on $F_1 , F_2 , ..... , F_{mkq}$ where J is a composition of k disjoint cycles of length mqJ a petal based at $\zeta_j$ maps under R to a petal based at $\zeta_{j+1}$

#### Petal theorem :

As there are $K_j$ such cycles of components for the rationally indifferent cycle $c_j$ , we see that there are at least $\sum_{j}$ critical points of P in $C$ thus $\sum k_j \leq d-1 \Rightarrow$ we can take the uppper bound to be $N(d-1)$

Make sure you visit the Arithmetical Dynamics Part 2 post of this Series before the Arithmetical Dynamics Part 3.

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