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

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

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