# Questions tagged [arithmetic-dynamics]

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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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### 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: $\clubsuit)a_{g}(1)=g(x)$ $\clubsuit)a_{g}(r)=g(a_{g}(r-1))$ ...
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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. ...
260 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 ...
453 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 ...
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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 ...
377 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 ...
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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 ...
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 ...