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.

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.

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

We are here with the Part 0 of the Arithmetical Dynamics Series. Let's get started....

Rational function \( R(z)= \frac {P(z)}{Q(z)} \) ; where P and Q are polynimials . There are some theory about fixed points .

Theorem:

Let \( \rho \) be the fixed point of the maps R and g be the Mobius map . Then \( gRg^{-1} \) has the same number of fixed points at \( g(\rho) \) as \( R \) has at \( \rho \).

Theorem :

If \( d \geq 1 \) , a rational map of degree d has previously \( d+1 \) fixed points in .

To each fixed point \( \rho \) of a rational maps R , we associate a complex number which we call the multiplier \( m(R , \rho) \) of R at \( \rho \) .

$$ m(r, \rho) = \{ R^{,}(\rho) ; \ if \ \rho \neq \propto \ and \ \frac {1}{R^{,}(\rho)} ; \ if \ \rho = \propto $$

Now, we dive into classification of fixed points and this is purely local matter , it applies to any analytic function and in particular , to the local inverse(when it exists ) of a rational map .

Make sure you visit the Introduction to Arithmetical Dynamics post of this Series.

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 $m\left|N \Rightarrow\left(P^{m}(z)-z\right)\right|\left(P^{N}(z)-z\right)$

The proof of the theorem in Part 0 :

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.

We are here with the Part 3 of the Arithmetical Dynamics Series. Let's get started....

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

Theory:

Let \( \{ \zeta_1 , ......., \zeta_m \} \) be a ratinally indifferent cycle for R and let the multiplier of \( R^m \) at each point of the cycle be \( exp \frac {2 \pi i r}{q} \) where \( (r,q) =1 \) . Then \( \exists \ k \in Z \) and\( mkq \) distinct component \( F_1 , F_2 , ..... , F_{mkq} \) s.t. at each \( \zeta_j \) there are exactly \( kq \) of these component containing a petal of angle \( \frac {2 \pi}{kq} \ at \ \zeta \) .

Further R acts as a permutation J on \( F_1 , F_2 , ..... , F_{mkq} \) where J is a composition of k disjoint cycles of length mqJ a petal based at \( \zeta_j \) maps under R to a petal based at \( \zeta_{j+1} \)

Petal theorem :

As there are \( K_j \) such cycles of components for the rationally indifferent cycle \( c_j \) , we see that there are at least \( \sum_{j} \) critical points of P in \( C \) thus \( \sum k_j \leq d-1 \Rightarrow \) we can take the uppper bound to be \( N(d-1) \)

Make sure you visit the Arithmetical Dynamics Part 2 post of this Series before the Arithmetical Dynamics Part 3.

We are here with the Part 2 of the Arithmetical Dynamics Series. Let's get started....

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

The lower bound calculation is easy .

But for the upper bound , observe that each \( z \in K \) lies in some cycle of length m(z) and we these cycles by \( C_1 , C_2 .....,C_q \) . Further , we denote the length of the cycle by \( m_j \) , so , if \( z \in C_j \) then , \( m(z)= m_j \) $$ \sum_{j=1}^{q} \sum_{z \in C_j} [\mu(N,z) - \mu(m_j ,z)] $$ we can confine our attention xparatly .

Now , \( \mu(N,z) = \mu(m_j -, z) \) whenever \( z \in C_j \rightarrow \) rationally indifferent .

So , nonzero contribution comes from rationally different cycles , \( C_j \) .

Theorem:

1. If m|n , then \( R^n \) has no fixed at \( \zeta_j \) .
   1. If m|n but \( m_q \not | n \) , then \( R^n \) has our fixed point at \( \zeta \) .
   2. If \( m_q |n \) then \( R^n \) has fixed point .

Make sure you visit the Arithmetical Dynamics Part 1 post of this Series before the Arithmetical Dynamics Part 2.

We are here with the Part 1 of the Arithmetical Dynamics Series. Let's get started....

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

Definition:

Suppose that \( \zeta \in C \) is a fixed point of an analytic function \( f \) .

Then \( \zeta \) is :

a) Super attracting if \( f^{'} (\zeta) =0 \rightarrow \) critical point of \( f \)

b) Attractting if \( 0 < |f^{'}( \zeta )|< 1 \ \rightarrow \) not a critical point of \( f \)

c) Repelling if \( |f^{'}( \zeta )|>1 \)

d)Rationally indifferent if \( f^{'}( \zeta ) \) is a root of unity .

e) Irratinally indifferent if \( |f^{' }( \zeta)|=1 \) , but \( f^{'}( \zeta ) \) is not a root of unity .

R has a period n ; \( R ^ {n} ( \zeta ) = \zeta \) .

If we denote \( R^m(\zeta) = \zeta-m ; m= 0, 1 ,2 ,3 ....... \) .

So $$ \zeta_{m+n}= \zeta_m \ then \ \ (R^n)^{'}( \zeta )= \prod_{i=0}^{n-1} ( \zeta_k ) \ [fixed] $$

So , we can say about attractuing , sup-attracting , repelling of \( R^n \) in terms of multiplier of \( R^n \) .

(Super)attracting points (cycles) relate to Faton set .

(Repelling) points (cycles) relate to pulin set .

Make sure you visit the Arithmetical Dynamics Introduction post of this Series before the Arithmetical Dynamics Part 1.