# Questions tagged [arithmetic-dynamics]

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

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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 $\alpha, \phi(\... 1answer 47 views ### Height argument for post critically finite$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, ... 1answer 473 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 ... 1answer 142 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 $$\{0, 0^2 + c, c^2 + ... 1answer 183 views ### Why Study Critical Polynomials? In dynamical systems, I often read about the post-critical orbits. As in take a moduli space of functions f which are self maps. Find general critical points, and see where they orbit. They would ... 1answer 118 views ### Dynamics: circle diffeo with an irrational rotation number and its rational approximation I need some help in proving the following that arises from the proof of the Denjoy's theorem: Let f: [0,1] \to [0,1] be an orientation preserving circle diffeomorphism topologically conjugated to ... 1answer 154 views ### Sharkovsky theorem and proving existence of a period three orbit Can you help me to prove this proposition? We saw this after having proved the Sharkovsky theorem. f : [0, 1] \mapsto [0, 1] continuous has a period four orbit: \left\{x_1 < x_2 <x_3 < ... 2answers 302 views ### Measure-preserving mapping Let (X, \mu, T) be a mesure-preserving mapping. Let A \subset X be a measurable subset such that any point in A eventually comes back to A. We define space (A, \mu_A), \mu_A ( B) = \mu (B) ... 1answer 134 views ### Measure on torus invariant under multiplication Let T: [0,1] \rightarrow [0,1] be a multiplication by \beta >1 mod 1. Show that h(x) d x is T-invariant where$$h (x) = \sum_{n \geq 0} \beta^{-n} \chi _{[0,T^n (1)]} (x)$$(\chi is ... 2answers 307 views ### Behavior of a Collatz-like mod-4 sequence: Do some numbers increase without limit? Define \begin{eqnarray} f(n) &=& (n-1)^2 \; \textrm{if} \; (n \bmod 4) = 1\\ f(n) &=& \lfloor n/4 \rfloor \; \textrm{otherwise} \end{eqnarray} and let f^k(n) = f(f( \cdots (n) \cdots )... 1answer 173 views ### A rational orbit that's provably dense in the reals? Iterating the map \ \ x\ \mapsto\ x-\frac{1}{x},\ \ the orbit of initial point 2 is "probably" dense in \mathbb{R}. Is there an explicit rational mapping together with an initial rational ... 0answers 74 views ### About Network Dynamics Consider the 6-node ring with only one bit active (000001). This is shown in the following figure as a hexagon with one circle filled. If the active bit is traveling in the counter clockwise ... 1answer 171 views ### Polynomial iterates with finite/infinite orbits Given a polynomial f\in\mathbb{C}[x] and a point x_0\in\mathbb{C} I am trying to decide if the set \{x_0,x_1=f(x_0),x_2=f(x_1),\cdots\} is finite or infinite. My general strategy is to iterate ... 0answers 61 views ### Looking for articles on postcritically finite rational maps in Russian or French I'm looking for articles on postcritically finite rational maps. I found a few articles in English, but I can't find any in Russian or French. 1answer 184 views ### Divisibility of an expression involving the Möbius function Let d and n be integers, with n\geq 2, and define$$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 ...