# Questions tagged [arithmetic-dynamics]

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

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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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Prove that $\displaystyle (\phi^n )' (\alpha) = \prod_{ i=0}^{n-1}\phi'( \phi^i (\alpha))$ . In particular, $\alpha$ is a critical point of $\phi^n$ if and only if one of the points $\alpha, \phi(\... 2 votes 1 answer 71 views ### Height argument for post critically finite$z^2+c$I know that if we require$c$to be rational the only post-critically finite maps of the form$z^2+c$have$c = 0$or$c=-1$. Is there a height argument for why this is true? It seems there must be, ... 7 votes 1 answer 631 views ### Crazy patterns arising from recursive sequence of functions (It should first be stated that I'm all new to this kind of stuff, so do tell if anything turns out to be obvious to the more experienced reader, or even incorrect.) I've been considering the ... 4 votes 1 answer 174 views ### Critical polynomial roots bigger than 2 In the question Why study critical polynomials?, it was said that each hyperbolic component of the Mandelbrot set contains a root for a critical polynomial. That is, the sequence $$\{0, 0^2 + c, c^2 + ... 7 votes 1 answer 269 views ### Why Study Critical Polynomials? In dynamical systems, I often read about the post-critical orbits. As in take a moduli space of functions f which are self maps. Find general critical points, and see where they orbit. They would ... 0 votes 1 answer 211 views ### Dynamics: circle diffeo with an irrational rotation number and its rational approximation I need some help in proving the following that arises from the proof of the Denjoy's theorem: Let f: [0,1] \to [0,1] be an orientation preserving circle diffeomorphism topologically conjugated to ... 1 vote 1 answer 212 views ### Sharkovsky theorem and proving existence of a period three orbit Can you help me to prove this proposition? We saw this after having proved the Sharkovsky theorem. f : [0, 1] \mapsto [0, 1] continuous has a period four orbit: \left\{x_1 < x_2 <x_3 < ... 0 votes 2 answers 504 views ### Measure-preserving mapping Let (X, \mu, T) be a mesure-preserving mapping. Let A \subset X be a measurable subset such that any point in A eventually comes back to A. We define space (A, \mu_A), \mu_A ( B) = \mu (B) ... 0 votes 1 answer 290 views ### Measure on torus invariant under multiplication Let T: [0,1] \rightarrow [0,1] be a multiplication by \beta >1 mod 1. Show that h(x) d x is T-invariant where$$h (x) = \sum_{n \geq 0} \beta^{-n} \chi _{[0,T^n (1)]} (x)$$(\chi is ... 6 votes 2 answers 353 views ### Behavior of a Collatz-like mod-4 sequence: Do some numbers increase without limit? Define \begin{eqnarray} f(n) &=& (n-1)^2 \; \textrm{if} \; (n \bmod 4) = 1\\ f(n) &=& \lfloor n/4 \rfloor \; \textrm{otherwise} \end{eqnarray} and let f^k(n) = f(f( \cdots (n) \cdots )... 4 votes 1 answer 222 views ### A rational orbit that's provably dense in the reals? Iterating the map \ \ x\ \mapsto\ x-\frac{1}{x},\ \ the orbit of initial point 2 is "probably" dense in \mathbb{R}. Is there an explicit rational mapping together with an initial rational ... 2 votes 0 answers 77 views ### About Network Dynamics Consider the 6-node ring with only one bit active (000001). This is shown in the following figure as a hexagon with one circle filled. If the active bit is traveling in the counter clockwise ... 1 vote 1 answer 250 views ### Polynomial iterates with finite/infinite orbits Given a polynomial f\in\mathbb{C}[x] and a point x_0\in\mathbb{C} I am trying to decide if the set \{x_0,x_1=f(x_0),x_2=f(x_1),\cdots\} is finite or infinite. My general strategy is to iterate ... 2 votes 0 answers 66 views ### Looking for articles on postcritically finite rational maps in Russian or French I'm looking for articles on postcritically finite rational maps. I found a few articles in English, but I can't find any in Russian or French. 4 votes 1 answer 243 views ### Divisibility of an expression involving the Möbius function Let d and n be integers, with n\geq 2, and define$$R(n,d) := \sum_{k\mid n} \mu( n/k)d^k,$$where$\mu$is the Möbius function. It can be shown (see below for an arithmetic proof) that$n\mid R(...
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 ...