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

Again, we are here with the Part 6 of the Arithmetical Dynamics Series. Let’s get started….

Consider fix point of $R(z) = z^2 – z$ .

Which is the solution of $$R(z) = z \\ \Rightarrow z^2 – z =z \\ \Rightarrow z^2 – 2z =0 \\ \Rightarrow z(z -2) =0$$

Now , consider the fix point of $R^2(z)$ . $\\$ $$i.e. R^2(z) = R . R(z) = R(z^2 -z)\\ \Rightarrow (z^2 -z)^2 – z^2 +z =z \\ \Rightarrow z^4 -2z^3 = 0 \\ \Rightarrow z^3(z- 2) =0$$

So , every solution of $R^2(z) =z$ is asolution of $R(z) =z$ .

Here comes the question of existence of periodic point .

I. N . Baker proved that ,

#### Theorem:

Let P be a polynomial of degree at least 2 and suppose that P has no periodic points of period n . Then n=2 and P is to $z \rightarrow z^2 – z$ .

#### Theorem:

Let $R , \ \ (\frac {P}{Q})$ be a rational function of degree $$d = max \{ degree(P) , degree(Q) \} , \ where \ d \geq 2 .$$

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