Related papers: A note on Misiurewicz polynomials

We examine the Galois groups of the extensions $K((f'\circ f^n)^{-1}(0))/K$ where $K$ is a number field for polynomials $f(x)\in K[x]$. We use our understanding of this group to study the proportion of primes for which $f$ has a $\mathfrak…

In this follow-up paper, we again inspect a surprising relationship between the set of $m$-periodic points of a polynomial map $\varphi_{d, c}$ defined by $\varphi_{d, c}(z) = z^d + c$ for all $c, z \in \mathcal{O}_{K}$ and the coefficient…

The multiplier $\lambda_n$ of a periodic orbit of period $n$ can be viewed as a (multiple-valued) algebraic function on the space of all complex quadratic polynomials $p_c(z)=z^2+c$. We provide a numerical algorithm for computing critical…

It is proved that for every complex quadratic polynomial $f$ with Cremer's fixed point $z_0$ (or periodic orbit) for every $\delta>0$, there is at most one periodic orbit of minimal period $n$ for all $n$ large enough, entirely in the disc…

We prove that any unicritical polynomial $f_c:z\mapsto z^d+c$ which is at most finitely renormalizable and has only repelling periodic points is combinatorially rigid. It implies that the connectedness locus (the ``Multibrot set'') is…

Let $K$ be a function field of a curve in characteristic zero or a number field over which the $abc$-conjecture holds, fix $\alpha\in K$, and let $f_{d,c}(x)=x^d+c$ for some $d\geq2$ and some $c\in K$. Then for many $c$ and $d$, we prove…

Given a number field $K$ and a polynomial $f(z) \in K[z]$, one can naturally construct a finite directed graph $G(f,K)$ whose vertices are the $K$-rational preperiodic points of $f$, with an edge $\alpha \to \beta$ if and only if $f(\alpha)…

An endomorphism $f:\mathbb{P}^k\to\mathbb{P}^k$ of degree $d\geq2$ is said to be postcritically finite (or PCF) if its critical set $\mathrm{Crit}(f)$ is preperiodic, i.e. if there are integers $m>n\geq0$ such that…

We study the preperiodic dynatomic curves $\mathcal{X}\_{n,p}$, the closure of set of $(c,z)\in \C^2$ such that $z$ is a preperiodic point of $f\_c$ with preperiod $n$ and period $p$ ($n,p\geq1$). We prove that each $\mathcal{X}\_{n,p}$ has…

We study the discrete dynamics of standard (or left) polynomials $f(x)$ over division rings $D$. We define their fixed points to be the points $\lambda \in D$ for which $f^{\circ n}(\lambda)=\lambda$ for any $n \in \mathbb{N}$, where…

In this paper we study two questions related to exceptional behavior of preperiodic points of polynomials in $\mathbb{Q}[x]$. We show that for all $d\geq 2$, there exists a polynomial $f_d(x) \in \mathbb{Q}[x]$ with $2\leq \mathrm{deg}(f_d)…

We classify the pairs of polynomials $f,g \in \mathbb{C}[X]$ having orbits satisfying infinitely many multiplicative dependence relations, extending a result of Ghioca, Tucker and Zieve. Moreover, we show that given $f_1,\ldots, f_n$ from a…

We provide the first definition of \emph{Misiurewicz parameter} for the unicritical family of algebraic correspondences $ z^r + c$, with $ r > 1$ rational, and prove that, at every Misiurewicz parameter, the correspondence uniformly expands…

For rational numbers $c$, we present a trichotomy of the set of totally real (totally $p$-adic, respectively) preperiodic points for maps in the quadratic unicritical family $f_c(x)=x^2+c$. As a consequence, we classify quadratic…

Let $c(x_1,...,x_d)$ be a multihomogeneous central polynomial for the $n\times n$ matrix algebra $M_n(K)$ over an infinite field $K$ of positive characteristic $p$. We show that there exists a multihomogeneous polynomial $c_0(x_1,...,x_d)$…

Let $K$ be a function field over an algebraically closed field $k$ of characteristic $0$, let $\varphi\in K(z)$ be a rational function of degree at least equal to $2$ for which there is no point at which $\varphi$ is totally ramified, and…

Given two rational maps $f,g: \mathbb{P}^1 \to \mathbb{P}^1$ of degree $d$ over $\mathbb{C}$, DeMarco-Krieger-Ye [DKY22] has conjectured that there should be a uniform bound $B = B(d) > 0$ such that either they have at most $B$ common…

Andrews and Petsche proposed in 2020 a conjectural characterization of all pairs $(f,\alpha)$, where $f$ is a polynomial over a number field $K$ and $\alpha\in K$, such that the dynamical Galois group of the pair $(f,\alpha)$ is abelian. In…

Let $d>k$ be positive integers. Motivated by an earlier result of Bugeaud and Nguyen, we let $E_{k,d}$ be the set of $(c_1,\ldots,c_k)\in\mathbb{R}_{\geq 0}^k$ such that…

Let $k$ be a number field with algebraic closure $\bar{k}$, and let $S$ be a finite set of places of $k$ containing all the archimedean ones. Fix $d\geq 2$ and $\alpha \in \bar{k}$ such that the map $z\mapsto z^d+\alpha$ is not…