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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 , \( C_j \) .

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

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