Arithmetical Dynamics: Part 5

Arithmetical dynamics is the combination of dynamical systems and number theory in mathematics. The basic objective of Arithmetical dynamics is to explain the arithmetic properties with regard to underlying geometry structures.

Again, we are here with the Part 5 of the Arithmetical Dynamics Series. Let's get started....

And suppose that R has no periodic points of period n . Then (d, n) is one of the pairs \( (2,2) ,(2,3) ,(3,2) ,(4 ,2) \) , each such pair does arise from some R in this way .

The example of such pair is $$ 1. R(z) = z +\frac {(w-1)(z^2 -1)}{2z} ; it \ has \ no \ points \ of \ period \ 3 .$$

$$ 2. If R(z) = \frac {z^3 +6}{3z^2} , then \ R \ has \ no \ points \ of \ period \ 2 \\ \\ \\ \\ R^2(z) = z \Rightarrow \frac {(\frac {z^3 +6}{3z^2})^3 + 6}{ 3 (\frac {z^3 + 6}{3z^2})^2 } =z \Rightarrow \frac {(z^3 +6)^3+(27 \times 6) z^6}{z^2 \times 3^2 \times (z^3 + 6)^2} =z $$

$$ 3. \ If \ R(z) = \frac {-z(1+ 2z^3)}{1-3z^3} \ then \ R \ no \ points \ of \ period \ 2 .$$

Make sure you visit the previous part of this Arithmetical Dynamics Series.