Arithmetical Dynamics: Part 2

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.

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