# Questions tagged [arithmetic-dynamics]

Arithmetic dynamics combines the study of dynamical systems with number theory.

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### Why are rotation numbers not homomorphic?

If $f,g$ are degree-1 monotone maps of the circle, why do we generally have $\rho(f\circ g)\neq\rho(f)+\rho(g)$? I mean, you might say that we have no right to expect an equality. After all, it's not ...
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### What are the prerequisites for learning Arithmetic dynamics?

I recently attended a webinar where the speaker was introducing some research topics to the audience. There I was introduced to Arithmetic Dynamics and found it to be an interesting topic. I wish to ...
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### To find an isomorphism from a hyperplane in an ambient projective space to another projective space?

Consider the hyperplane $$X:=V(a_0x_0+a_1x_1+\cdots+a_Nx_N+a_{N+1}x_{N+1}) \subset \mathbb P^{N+1},$$ where $a_0,a_1, \cdots, a_{N+1} \in \mathbb Q$ are not all zero. Define the height function as ...
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### In the iteration defining the arithmetic-geometric mean, how many terms of both sequences can be integers?

The definition of the arithmetic-geometric mean ($\text{AGM}$) for two numbers $a_0$ and $b_0$ is to define two sequences $a_n$ and $b_n$ as $a_{n+1}=\frac{a_n+b_n}{2}$ and $b_{n+1}=\sqrt{a_n b_n}$, ...
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### What are some applications of arithmetic dynamics?

In classical real or complex dynamics, we iterate over the reals or complex numbers. One application of this, among many, is the discrete logistic map for population growth. In arithmetic dynamics, we ...
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### Preperiodic point and Group Torsion

Let $G$ be a group, let $d \geq 2$ be an integer, and define a map $\phi(g)=g^d$, Prove that $PrePer(\phi, G)= G_{tors}$, i.e., prove that the preperiodic points are exactly the points of finite order ...
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### What is exactly "Algebraic Dynamics"?

Could somebody please give a big picture description of what exactly is the object of study in the area of Algebraic Dynamics? Is it related to Dynamical Systems? If yes in what sense? Also, what is ...
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In short, I need to prove whether I have preperiodic orbits or wandering orbits for some dynamical systems. Not as short: Let $\phi$ be a rational map. Suppose we want to know if 0 is preperiodic ...