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