We are here with the Part 4 of the Arithmetical Dynamics Series. Let's get started....
Arithmetical dynamics is the combination of dynamical systems and number theory in mathematics.
$P^{m}(z)=z$ and $P^{N}(z)=z$ where $m\left|N \Rightarrow\left(P^{m}(z)-z\right)\right|\left(P^{N}(z)-z\right)$
Let , P be the polynomials satisfying the hypothesis of theorem 6.2.1 .
Let, $K=\left\{z \in C \mid P^{N}(z)=z\right\}$
and let $M={m \in Z: 1 \leq m \leq N, m \mid N}$
then each $z \in K$ is a fixed point of $P^{m}$ for some $m \in M$ and we let $\mathrm{m}(\mathrm{z})$ be the minimal such $\mathrm{m}$.
The proof depends on establishing the inequalities ,
$d^{N-1}(d-1)$ $\leq \sum_{k}[\mu(N, z)-\mu(m(z), z)]$ $\leq N(d-1)$ ....... (1)
where $\mu(n, w)$ is the no of fixed points of $P^{n}$ at w.
$(1) \Rightarrow d^{N-1} \leq N$
therefore,
$N$=$1+(N-1)$ $\leq 1+(N-1)(d-1)$ $\leq[1+(d-1)]^{N-1}$=$d^{N-1}$ $\leq N$
Make sure you visit the Arithmetical Dynamics Part 3 post of this Series before the Arithmetical Dynamics Part 4.
