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Questions tagged [arithmetic-dynamics]

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

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Evaluate the change in the potential energy

If the force $$F=3i +4j $$ moves a body from the position $$A(2 , 3)\ \mathrm{to}\ B(7 , 6) $$ .Evaluate the change in the potential energy of the body . My turn: The work done by the force =$$(3,4) •...
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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 ...
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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 ...
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How are the values for velocity and mass found?

I'm confused as to how and where the solutions have derived the mass and velocity of A and B in this question. If someone could help me understand, that would be great.
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Determining distance between two distances given acceleration, max velocity, deceleration and total time.

PROBLEM: A train accelerates from rest at one train station with $a= 0.72t$ $m/s^2$, until it reaches its maximum speed of $18 m/s$. Its deceleration heading into the next train station is $3.0 m/s^2$....
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49 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}={\...
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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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130 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 + ...
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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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Position of a particle, Newtons Second Law

A particle of mass $40$ kg moves in a straight line such that the force (in newtons) acting on it at time $t$ (in seconds) is given by $$160t^4-320t^2-360$$ If at time $t=0$ its velocity $v$ is given ...
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How can the reaction force of an object be exerted on both bodies?

Assume you have 2 bodies A on top of B. They're both accelerating upwards at 0.5 ms^-2. This is my logic for finding the force exerted on B by A (correct me if I'm wrong) I was told that to find this,...
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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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Can someone clear up this statement on resisted motion (Dynamics)?

Due to missing a lecture (for health reasons), I'm reading my lecturer's notes on the resisted motion of a particle where air resistance is proportional to the square of the particle's speed. Here is ...
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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 ...