Questions tagged [arithmetic-dynamics]

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

Filter by
Sorted by
Tagged with
0
votes
0answers
24 views

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 $...
2
votes
0answers
14 views

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 ...
1
vote
1answer
58 views

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 ...
2
votes
0answers
79 views

$||g_n||_{\infty} < \delta_{n-1}(g)$

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 $|\lambda| = \max_{1 \leq j \leq d}|\lambda_j|$ ...
2
votes
1answer
55 views

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 ...
2
votes
1answer
80 views

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 : \...
1
vote
1answer
30 views

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 ...
2
votes
0answers
32 views

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},...
0
votes
0answers
41 views

Can I get a map $f: \Bbb P^2 \to \Bbb P^2$ birational such that $fσ=σf$?

Consider the The standard quadratic involution.$ σ: \Bbb P^2 \to \Bbb P^2$ defined by $[z_0, z_1, z_2] → [1/z_0, 1/z_1, 1/z_2] = [z_1z_2, z_0z_2, z_0z_1]$. Can I get a map $f: \Bbb P^2 \to \Bbb P^2$ ...
3
votes
0answers
191 views

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 ...
4
votes
1answer
84 views

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}^...
1
vote
1answer
51 views

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(\...
1
vote
2answers
110 views

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 ...
0
votes
0answers
39 views

Plotting complicated function on sagemath

I am not sure if this is the right place to ask about coding a program sagemath. But it is the only math online community I know, so I hope I can get some suggestions here. The problem is I want to ...
1
vote
0answers
59 views

Siegel Transforms on Homogeneous Spaces?

For any integer $n \geq 2,$ we may identify the space of unimodular lattices in $\mathbb{R}^n$ with the homogeneous space $X_n := \mathrm{SL}_n(\mathbb{R})/\mathrm{SL}_n(\mathbb{Z})$ via the ...
0
votes
1answer
127 views

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}={\...
1
vote
0answers
32 views

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(\...
2
votes
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, ...
7
votes
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 ...
4
votes
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 + ...
7
votes
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 ...
0
votes
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 ...
0
votes
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 < ...
0
votes
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) ...
0
votes
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 ...
6
votes
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 )...
4
votes
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 ...
2
votes
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 ...
1
vote
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 $...
2
votes
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.
4
votes
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(...
8
votes
2answers
1k views

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 ...
3
votes
2answers
273 views

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 ...