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 ...
Ajin Shaji Jose's user avatar
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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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Preperiodic points and finite orbits

Let $X$ be a set and $f:X\to X$ a map. An element $x\in X$ is said to be preperiodic if $\exists n>m\geq 0$ such that $f^{n+m}(x)=f^m(x)$, where $f^k(x):=\underbrace{f\circ\cdots\circ f}_{\text{$k$-...
mahlovic's user avatar
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The failure of Arzela-Ascoli over non-archimedean fields

Let $K$ be a complete, algebraically closed non-archimedean field. The Arzela-Ascoli theorem fails over such a field $K$. I found an example on a book, but not quite get it. Here is an example : $\...
Nothingone's user avatar
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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 ...
J W's user avatar
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Orbits of $f(z) = 2z^3+z.$

Let $f(z) = 2z^3+z.$ What are the values of $z$ such that $z, f(z), f(f(z)), \dots$ is eventually periodic? I'm asking because I wish to classify all monic polynomials $p \in \mathbb{Z}[x]$ such that $...
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While solving the motion in plane problems (dynamics) how to figure out whether the radial accelration is 0 or mgsin(theta)?

There is a problem : A straight smooth tube revolves with constant angular velocity W in a horizontal plane about one extremity which is fixed. If at zero time the tube be horizonal and a particle ...
Soumantrik Ganguly's user avatar
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does a slight change in a real value cause a massive change from finite to infinite?

Can anybody give an example of a system of equivalence and/or inequality relations dependent to just a real value number like $\kappa$, which satisfies for finitely many integers if $\kappa =1$ and ...
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$||g_n||_{\infty} < \delta_{n-1}(g)$

Let $E$ be a (possibly infinite) alphabet and consider $X = E^{\mathbb{Z}^d}$. For each $\lambda = (\lambda_1, \dots, \lambda_d) \in \mathbb{Z}^d$, let $|\lambda| = \max_{1 \leq j \leq d}|\lambda_j|$ ...
Luísa Borsato's user avatar
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Accumulation point for closure wrt two particular contracting affine maps on $(-1,1)$

Let $B(x)=\frac{3x+1}{4}$ and $C(x)=\frac{3x-1}{4}$. Then $(-1,1)$ is a stable interval for both $B$ and $C$. Let $S\subseteq (-1,1)$ be nonempty and stable by both $B$ and $C$. Note that for any ...
Ewan Delanoy's user avatar
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What is the topological entropy of the Collatz map (extended to 2-adic integers)?

In the process of learning the basics of dynamical systems, I finished the chapter on topological entropy and decided, as an exercise, to try and compute the entropy of one of my favorite maps: $T : \...
Rob's user avatar
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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 ...
julioprofe's user avatar
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Can I get a map $f: \Bbb P^2 \to \Bbb P^2$ defined by the monomial dehomogenization $[z_1, z_2] → [z_1^2z_2, z_1z_2]$ birational such that $fσ=σf$?

Consider the The standard quadratic involution.$ σ: \Bbb P^2 \to \Bbb P^2$ defined by the monomial dehomogenization $[z_1, z_2] → [z_1^2z_2, z_1z_2]$ which has the inverse $[z_1, z_2] → [z_1z_2^{-1},...
Baby Elephant's user avatar
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Some extended question on bounds of Rational map

I saw this question here: I am stuck on the same kind of question but my problem is a bit more general which thrives me to post a new one. I am copying a bit definition from that post to save some ...
Ri-Li's user avatar
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Finding an upper bound to the order of finite subgroup of the automorphism group of rational map

Let $\phi: \Bbb P^1 \to \Bbb P^1$ be a rational map then we define $Aut(\phi)=\{f \in PGL_2(\Bbb C): f^{-1}\phi f(z)=\phi(z)\}$ Here, in general, the definition of a rational map is: Let $\mathbb{P}^...
Baby Elephant's user avatar
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Problem 3.8 of Silverman's 'Arithmetic of Dynamical Systems'

Let $\phi \in K(z)$ be a rational function of degree $d$ over a number field $K$ and write $\phi = f/g$ for coprime $f,g \in K[z]$. Then, there exist positive constants $c_1, c_2$ such that $c_1 H(\...
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ODE with infinitely many solutions [duplicate]

We have that: $ \dot{x}=x^{2/3} $, and $x(0)=0$. This Initial Value Problem has infinitely many solutions (is a problem in my Dynamics Book, so I'm sure about that), but personally I could found only ...
arthll's user avatar
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Every quadratic polynomial is conjugate to one of the polynomial $z^2+t$

Every quadratic polynomial is conjugate to one of the polynomial $z^2+t$ where conjugation means conjugation by a linear fractional transformation $\alpha(z)=\frac{az+b}{cz+d}$ and $f^{\alpha}={\...
Ri-Li's user avatar
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Prove that $(ϕ^n )' (α) = \prod_{ i=0}^{n-1}ϕ'( ϕ^i (α)) $

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(\...
Ri-Li's user avatar
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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, ...
Indy500's user avatar
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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 ...
giobrach's user avatar
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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 + ...
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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 ...
mdave16's user avatar
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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 ...
pray's user avatar
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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 < ...
pray's user avatar
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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) ...
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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 ...
wroobell's user avatar
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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 )...
Joseph O'Rourke's user avatar
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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 ...
r.e.s.'s user avatar
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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 ...
Philip Benj's user avatar
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1 answer
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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 $...
Charles's user avatar
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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.
Jacques2013's user avatar
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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(...
froggie's user avatar
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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 ...
Manos's user avatar
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Detecting preperiodic points

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 ...
Alex's user avatar
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