# Questions tagged [arithmetic-dynamics]

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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 $f:\mathbb{N}\to\mathbb{N}$ ...
1answer
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. ...
0answers
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 $\{a_i\} \in D(a,r)$ ...
0answers
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 ...
0answers
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 ...
2answers
1k views

2answers
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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 ...
2answers
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....
0answers
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 $t_1$...
1answer
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 ...
0answers
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 ...
1answer
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 ...
1answer
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 ...
2answers
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 ...
5answers
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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 ...
0answers
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 ...
0answers
481 views