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.
Publishing original scientific research in a peer-reviewed journal is an achievement many scholars work toward for years. Achieving this feat early in one's academic journey requires exceptional intellectual rigor, creativity, and persistence. Aaditya Punatar and Muskan Sadana have reached this remarkable milestone, successfully publishing their collaborative research paper in the prestigious Journal of Emerging Investigators […]
Her project on quantum-accelerated Steiner trees for 15-minute cities earned a place among the finalists at the world's largest pre-college STEM competition. Prishaa Shrimali of Marymount School of New York, was selected as a Grand Finalist at the 2026 Regeneron International Science and Engineering Fair (ISEF) in Phoenix, Arizona. Her project was Optimizing Urban Accessibility: […]
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's M.Stat Entrance. They ranked within the first 50 in the entire country in these entrances. I.S.I. M.Stat Entrance
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's B.Stat Entrance and Chennai Mathematical Institute's B.Sc. Math Entrance. They ranked within the first 200 in the entire country in these entrances. Most of these students attended the problem solving workshops regularly, which happen 5 days every week. CMI B.Sc. Math Entrance […]
