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6 votes
0 answers
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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 ...
Anwesh Ray's user avatar
0 votes
1 answer
83 views

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(...
ShyamalSayak's user avatar
3 votes
0 answers
181 views

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{\...
Superguy's user avatar
  • 356
4 votes
0 answers
328 views

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 (...
Asvin's user avatar
  • 7,716
4 votes
1 answer
394 views

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: $\clubsuit)a_{g}(1)=g(x)$ $\clubsuit)a_{g}(r)=g(a_{g}(r-1))$ ...
Hhhhhhhhhhh's user avatar
  • 1,032
2 votes
1 answer
207 views

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. ...
Hhhhhhhhhhh's user avatar
  • 1,032
27 votes
6 answers
2k views

$m$-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}...
jack's user avatar
  • 2,929
4 votes
1 answer
458 views

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 ...
nomadd's user avatar
  • 41
2 votes
0 answers
477 views

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 ...
MAS's user avatar
  • 922
8 votes
1 answer
323 views

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 \...
Sebastien Palcoux's user avatar
9 votes
0 answers
319 views

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 ...
Vesselin Dimitrov's user avatar
0 votes
1 answer
201 views

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 ...
user152801's user avatar
1 vote
1 answer
162 views

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 ...
Constantin K's user avatar
2 votes
0 answers
134 views

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 $f:\mathbb{N}\to\mathbb{N}$ ...
Harry Altman's user avatar
  • 2,585
2 votes
1 answer
476 views

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. ...
Ariyan Javanpeykar's user avatar
4 votes
0 answers
145 views

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 $\{a_i\} \in D(a,r)$ ...
Panna's user avatar
  • 61
3 votes
0 answers
295 views

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 ...
Boris's user avatar
  • 39
13 votes
0 answers
494 views

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 ...
Nikita Kalinin's user avatar
22 votes
2 answers
1k views

$x_1 = 2$, $x_{n + 1} = {{x_n(x_n + 1)}\over2}$, 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 ...
user avatar
10 votes
2 answers
427 views

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 ...
Lee Mosher's user avatar
  • 15.4k
5 votes
0 answers
215 views

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 ...
user avatar
7 votes
1 answer
914 views

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{...
Fedor Petrov's user avatar
11 votes
2 answers
564 views

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 ...
Gheehyun Nahm's user avatar
4 votes
2 answers
517 views

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....
Spock's user avatar
  • 443
1 vote
0 answers
237 views

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 $t_1$...
Jeremy K's user avatar
16 votes
1 answer
2k views

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 ...
bzm3r's user avatar
  • 398
3 votes
0 answers
132 views

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 ...
Subhajit Jana's user avatar
0 votes
1 answer
464 views

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 ...
wongpin101's user avatar
10 votes
1 answer
376 views

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 ...
joro's user avatar
  • 24.7k
14 votes
2 answers
746 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 ...
Sidney Raffer's user avatar
17 votes
5 answers
2k views

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 ...
user avatar
0 votes
0 answers
146 views

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 ...
Shanti's user avatar
  • 1
13 votes
0 answers
551 views

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^...
Lubin's user avatar
  • 4,078
4 votes
0 answers
329 views

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 ...
Mahdi Majidi-Zolbanin's user avatar
6 votes
4 answers
762 views

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 ...
Mahdi Majidi-Zolbanin's user avatar
11 votes
3 answers
1k views

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 ...
JSE's user avatar
  • 19.1k
12 votes
0 answers
392 views

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 ...
Joe Silverman's user avatar
26 votes
7 answers
2k views

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 ...
Mahdi Majidi-Zolbanin's user avatar
7 votes
2 answers
416 views

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" ...
JSE's user avatar
  • 19.1k
13 votes
1 answer
1k views

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 ...
Mark Dickinson's user avatar