# Questions tagged [arithmetic-dynamics]

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27
questions

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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**

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**1**answer

239 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. ...

**4**

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123 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 $\{a_i\} \in D(a,r)$ ...

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241 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 ...

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418 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 ...

**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 ...

**10**

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**2**answers

297 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 ...

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185 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 ...

**6**

votes

**1**answer

618 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{...

**11**

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**2**answers

464 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 ...

**3**

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**2**answers

326 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....

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**0**answers

225 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 $t_1$...

**15**

votes

**1**answer

875 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 ...

**3**

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**0**answers

121 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 ...

**0**

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**1**answer

218 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 ...

**10**

votes

**1**answer

310 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 ...

**14**

votes

**2**answers

608 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 ...

**16**

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**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 ...

**0**

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**0**answers

144 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 ...

**13**

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**0**answers

481 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^...

**4**

votes

**0**answers

309 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 ...

**6**

votes

**4**answers

655 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 ...

**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 ...

**12**

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**0**answers

370 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 ...

**24**

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 ...

**7**

votes

**2**answers

389 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" ...

**13**

votes

**1**answer

955 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 ...