Cheenta
How 9 Cheenta students ranked in top 100 in ISI and CMI Entrances?
Learn More

Arithmetical Dynamics: Part 6

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.

Knowledge Partner

Cheenta is a knowledge partner of Aditya Birla Education Academy
Cheenta

Cheenta Academy

Aditya Birla Education Academy

Aditya Birla Education Academy

Cheenta. Passion for Mathematics

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.
JOIN TRIAL
support@cheenta.com