Questions tagged [arithmetic-dynamics]
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40 questions
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Special value of the Artin--Mazur zeta function in arithmetic dynamics
Let $X$ be a compact manifold and $f: X\rightarrow X$ be a diffeomorphism. Assume that the $k$-fold iterate $f^k: X\rightarrow X$ has finitely many fixed points for all natural numbers $k$. The ...
- 685
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1
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89
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A question about the backward orbit and forward orbit of a rational map in 1-dimensional projective space
I am a bit puzzled about some probable implication in a paper.
Let $\varphi:\mathbb{P}^1_K\to \mathbb{P}^1_K$ be a rational map, where $K$ is a number field and let $\alpha\in K$ be such that $\varphi(...
user481562
3
votes
0
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183
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Stronger estimates for dynamical analogue of a sum of multiplicative order and primes
Let $a$ be a positive integer and $\mathrm{ord}_{p}(a)$ denote the multiplicative order of $a$ modulo $p.$ We know by a result of Murty, Silverman and Rosen that the sum $$\sum_{p~\text{prime}} \frac{\...
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Why are critical points important for dynamical systems?
I have just started reading a little about (arithmetic) dynamics and it seems like critical points are very important - for instance, rational maps so that critical points have finite forward orbit (...
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4
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1
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Is it true that sum of reciprocal of primes $p$ such that $p|a_{f}(p)$ converges?
Let $g(x)$ be a polynomial with integral coefficients.
For $r\geq 1$, We define the sequence $a_{g}$ for some polynomial $g(x)$ as follows:
$\clubsuit)a_{g}(1)=g(x)$
$\clubsuit)a_{g}(r)=g(a_{g}(r-1))$ ...
- 1,042
2
votes
1
answer
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If $a_{g}(1)=g(x)$ and $a_{g}(r)=g(a_{g}(r-1))$ for $r>1$ then is it true that $\limsup\limits_{r\to\infty}\gamma(a_{r})=\infty?$
Let $g(x)$ be a polynomial with integral coefficients.We define $\gamma(g(x))$ to be the degree of the non constant polynomial $r(x)$ which divides $g(x)$ for all $x$ and also has minimal degree.
...
- 1,042
29
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6
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$m$-fold composite $p^{(m)}(x) \in \mathbb{Z}[x]$ implies $p(x) \in \mathbb{Z}[x]$
Let $p(x)$ be a polynomial, $p(x) \in \mathbb{Q}[x]$, and $p^{(m+1)}(x)=p(p^{(m)}(x))$ for any positive integer $m$.
If $p^{(2)}(x) \in \mathbb{Z}[x]$ it's not possible to say that $p(x) \in \mathbb{Z}...
- 3,153
4
votes
1
answer
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All rational periodic points
I am trying to find all rational periodic points of a polynomial. To specify: a periodic point is the point that satisfy $f^n(x)=x$. It is related to dynamical systems in fact. So the current codes ...
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Is there bijective correspondence between $P_n$ and $A_n$?
Let $K \supset \mathbb{Q}_p$ be the $p$-adic field and let $O_K$ be its ring of integers and $M_K$ be the maximal ideal with integral closure $\bar{M}_K$. A power series is invertible if its lowest ...
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1
answer
328
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On the density map of the abundancy index
Let $σ$ be the sum-of-divisors function. Let $σ(n)/n$ be the abundancy index of $n$. Consider the density map $$f(x) = \lim_{N \to \infty} f_N(x) \ \ \text{ with } \ \ f_N(x) = \frac{1}{N} \#\{ 1 \...
9
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Discriminants of Gleason's period-$n$ polynomials for the Mandelbrot set
Gleason's polynomials are the sequence of monic integer polynomials defined recursively by
$$
\prod_{d \mid n} G_d(c) = (((c^2+c)^2+c)^2+\cdots+c)^2+c \quad \quad \quad [\textrm{$n$ iterates}],
$$
for ...
- 13.8k
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1
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Can the immediate basin of attraction of super-attracting fixed point at 0 of a polynomial contain non-zero roots?
Let $f$ be a polynomial with a super attracting fixed point at $x=0$. Can the immediate basin of attraction of the fixed point contain other roots? If so, please provide a specific example with the ...
1
vote
1
answer
162
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Why is the relative trace of Sobolev norms finite?
I am reading the 2009 Paper on Effective Equidistribution by Einsiedler, Margulis and Venkatesh (EMV). I do not understand Section 5.3 on the proof of (3.10). They want to prove that the relative ...
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0
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Can this construction generate bounded aperiodic functions?
This question is based on this old MathOverflow question: How this set of functions is ordered?
In that question, Vladimir Reshetnikov asked about a class $S$ of functions $f:\mathbb{N}\to\mathbb{N}$ ...
- 2,585
2
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1
answer
493
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Does Lang's conjecture imply Morton-Silverman's Uniform Boundedness conjecture?
I was curious to see whether the following conjecture of Morton-Silverman is (known to be) a consequence of Lang (or Lang-Vojta's) conjecture.
Conjecture. Let $D$, $N$, and $d$ be positive integers. ...
- 9,497
4
votes
0
answers
150
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The image of annuli of the non-Archimedean projective line by rational functions
I'm reading the book "potential theory and dynamics over the Berkovich projective line" by Baker and Rumely. The proposition 2.18 in this claims that if you choose suitable finite $\{a_i\} \in D(a,r)$ ...
- 61
3
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0
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301
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Paper by Moser on commuting circle diffeomorphisms and simultaneous Diophantine approximations
I am reading Moser's paper On commuting circle mappings and simultaneous Diophantine approximations and I found it hard because it is my first time that I seriously have to read a paper. It is a local ...
- 39
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0
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496
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Strange formula in arithmetic dynamic
Added: another function like that is $S_p f(z) = f(z)+\frac{f(\sqrt{zp})^2}{f(p)}$ in a field of characteristic two.
We discovered the following operator which acts on the space of polynomials (or ...
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votes
2
answers
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$x_1 = 2$, $x_{n + 1} = {{x_n(x_n + 1)}\over2}$, what can we say about $x_n \text{ mod }2$?
This question was asked on MathStackexchange here, but there was no answer, so I am asking it here.
Let$$x_1 = 2, \quad x_{n + 1} = {{x_n(x_n + 1)}\over2}.$$What can we say about the behavior of $x_n ...
user61522
10
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2
answers
439
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Growth of an integer vector under the action of a matrix in $GL_n(\mathbb{Z})$
I have some questions regarding the dynamics of elements of $GL_n(\mathbb{Z})$ acting on $\mathbb{Z}^n$. In particular, given an invertible integer matrix $M \in GL_n(\mathbb{Z})$, and given an ...
- 15.4k
5
votes
0
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216
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Dynamical Mordell-Lang on Kahler manifolds?
Suppose that $X$ is a smooth projective variety over $\mathbb C$ and $\phi : X \to X$ is an endomorphism. Let $p \in V$ be a point and $V \subset X$ a subvariety. The dynamical Mordell-lang ...
user47305
7
votes
1
answer
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Algebraic dynamics in finite fields
What is known about combinatorial structure of the rational maps of degree 2 over finite fields? From some general reasons I think it was studied. For being more specific, consider the field $\mathbb{...
- 109k
11
votes
2
answers
576
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A condition for a sequence defined by a recurrence relationship to all be integers
I am interested in a specific sequence $\{a_n \}$ defined by a simple recurrence relationship: $$a_n = \frac {a_{n-1} ^2 +c} {b} $$ where $b,c\in \mathbb{Z}$. I want to find all $b,c$ such that there ...
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4
votes
2
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526
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Growth of the size of iterated polynomials
I have been working independently on a project but now I am stuck and need to seek an expert's wisdom for a part of it. I am basically looking for theorems related to growth of the size of polynomials....
- 443
1
vote
0
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238
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A probability application question
Suppose there are two possible states $H$ and $L$, with prior probability $p$ and
$1-p$ respectively. There are infinite rounds with a discount factor $ d$. In
round 1, you could choose a value $t_1$...
- 11
16
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1
answer
2k
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Why is the Dynamical Mordell-Lang conjecture interesting?
The gist of the Dynamical Mordell-Lang conjecture is:
Let's say you have a set $S$. Inside $S$, you have a point $x$ where you start off at. Now, you also have a function $f: S \rightarrow S$ (is ...
- 398
3
votes
0
answers
132
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Equidistribution of double coset
Let $G=PGL_n(\mathbb{R})$, $K=PO_n(\mathbb{R})$ and $X=G/K$. Also suppose $\Gamma=SL_n(\mathbb{Z})$ acts on the left of $X$. We define a typical Hecke operator on $L^2(\Gamma\backslash X)$ by the ...
- 1,692
0
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1
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486
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Indeterminancy locus of rational maps
Let $K=\bar{\mathbb{Q}}(\mathbb{P}^2_\bar{\mathbb{Q}})$, the function field of $\mathbb{P}^2_\bar{\mathbb{Q}}$. Let $C/K$ be a smooth projective curve over $K$ in $\mathbb{P}^2_K$ and let $f$ be a ...
- 61
10
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1
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Reducibility of polynomials maps
Motivated by this question.
Let $f \in \mathbb{Q}[x]$ or$f \in \mathbb{Z}[x]$ .
Consider the sequence $f(x),f(f(x)), \ldots f^n(x)$.
If some $f^k(x)$ is reducible, the rest iterates will be ...
- 25.4k
14
votes
2
answers
756
views
Generating primes via composition of polynomials
It is well known that no nonconstant polynomial $f\in \mathbb{Z}[x]$ can assume only prime values at integer arguments. Indeed, if $a\in \mathbb{Z}$ is so large that $|f(a)|>1$, and if $p$ is a ...
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17
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5
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Arithmetic dynamics and dynamics on moduli spaces
The following question is more of a request for pointers to suitable literature on introductory material for arithmetic dynamics and dynamics on moduli spaces.
In my dissertation, I have been ...
user39719
0
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0
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cat map re-transformation
Hi,
Is there any way of moving from one cat map transformation to the other without resetting parameters?
For example, suppose you have two matrices '$A$'and '$B$' each permuted with different cat ...
- 1
13
votes
0
answers
556
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Higher-dimensional algebraic subgroups of the proalgebraic Nottingham group?
Let $R$ be a commutative ring, and, for $n\ge0$,
${\mathcal{A}}_n={\mathcal{A}}_n(R)$ the group of series
$u(x)=\sum_0^\infty a_jx^{j+1}\in R[[x]]$ for which
$a_0\in R^\times$ and $u(x)\equiv x\pmod{x^...
- 4,193
4
votes
0
answers
335
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Algebraic Dynamics over separated schemes
I have a few questions regarding the current status of research on algebraic dynamics over separated schemes. In what follows $\varphi:X\rightarrow X$ will be a finite self-morphism of a noetherian ...
- 3,101
6
votes
4
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763
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A follow up question related to entropy
For a self-map $\varphi:X\longrightarrow X$ of a space $X$, many important notions of entropy are defined through a limit of the form $$\lim_{n\rightarrow\infty}\frac{1}{n}\log a_n,$$ where in each ...
- 3,101
11
votes
3
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1k
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Greatest common divisor of a^{2^n}-1 and b^{2^n}-1
Let a and b be coprime integers. Do we know, expect, or unexpect that there are infinitely many primes p which divide
$gcd(a^{2^n} - 1, b^{2^n}-1)$
for some n? Certainly any Fermat prime will ...
- 19.2k
12
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0
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394
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Rational maps whose complex conjugate equals a PGL conjugate
Let $f(z)\in\mathbb{C}(z)$ be a rational function, and let $\bar{f}(z)$ denote the function obtained by taking the complex conjugate of the coefficients of $f$. I am interested in maps $f$ for which ...
- 47.4k
26
votes
7
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If you were to axiomatize the notion of entropy
What are the axioms that a good notion of entropy must satisfy? Please note that I am not asking for the definitions of various types of entropy such as topological entropy or measure-theoretic ...
- 3,101
7
votes
2
answers
417
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Dynamics of a random "quadratic" directed graph
Let G be a directed graph on N vertices chosen at random, conditional on the requirement that the out-degree of each vertex is 1 and the in-degree of each vertex is either 0 or 2. The "periodic" ...
- 19.2k
13
votes
1
answer
1k
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Conjectures on iterated polynomial maps on finite fields
Let $p$ be a prime, and consider the sequence $x_0, x_1, \dots$ of elements of the finite field $\mathbf F_p$ given by $x_0 = 0$ and $x_{i+1} = x_i^2 + 1$ for all $i \ge 0$. This sequence must ...
- 233