Related papers: Finite orbit points for sets of quadratic polynomi…

Let $S$ be the collection of quadratic polynomial maps, and degree $2$-rational maps whose automorphism groups are isomorphic to $C_2$ defined over the rational field. Assuming standard conjectures of Poonen and Manes on the period length…

Given a quadratic polynomial with rational coefficients, we investigate the existence of consecutive squares in the orbit of a rational point under the iteration of the polynomial. We display three different constructions of $1$-parameter…

The purpose of this note is give some evidence in support of conjectures of Poonen, and Morton and Silverman, on the periods of rational numbers under the iteration of quadratic polynomials. In particular, Poonen conjectured that there are…

We consider the polynomials $\displaystyle f(x)=x^d+c$, where $d\ge 2$ and $c\in\mathbb Q$. It is conjectured that if $d=2$, then $f$ has no rational periodic point of exact period $N\ge 4$. In this note, fixing some integer $d\ge 2$, we…

Bounding the number of preperiodic points of quadratic polynomials with rational coefficients is one case of the Uniform Boundedness Conjecture in arithmetic dynamics. Here, we provide a general framework that may reduce finding periodic…

Answering a question posed by Adam Epstein, we show that the collection of conjugacy classes of polynomials admitting a parabolic fixed point and at most one infinite critical orbit is a set of bounded height in the relevant moduli space.…

Let K be a number field, let f: P_1 --> P_1 be a nonconstant rational map of degree greater than 1, let S be a finite set of places of K, and suppose that u, w in P_1(K) are not preperiodic under f. We prove that the set of (m,n) in N^2…

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…

Let $K$ be a number field and $\phi\in K(z)$ a rational function. Let $S$ be the set of all archimedean places of $K$ and all non-archimedean places associated to the prime ideals of bad reduction for $\phi$. We prove an upper bound for…

Let $P\in\mathbb{P}_1(\mathbb{Q})$ be a periodic point for a monic polynomial with coefficients in $\mathbb{Z}$. With elementary techniques one sees that the minimal periodicity of $P$ is at most $2$. Recently we proved a generalization of…

Let f(z) be a rational function of degree at least 2 with coefficients in a number field K, and assume that the second iterate f^2(z) of f(z) is not a polynomial. The second author previously proved that for any b in K, the forward orbit…

It has been conjectured that for $N$ sufficiently large, there are no quadratic polynomials in $\bold Q[z]$ with rational periodic points of period $N$. Morton proved there were none with $N=4$, by showing that the genus~$2$ algebraic curve…

We classify the graphs that can occur as the graph of rational preperiodic points of a quadratic polynomial over $\bold Q$, assuming the conjecture that it is impossible to have rational points of period $4$ or higher. In particular, we…

We study rational periodic points of polynomial $f_{d,c}(x)=x^d+c$ over the field of rational numbers, where $d$ is an integer greater than 2. For period 2, we classify all possible periodic points for degrees $d=4,6$. We also demonstrate…

Over a number field $K$, a celebrated result of Silverman states that if $\varphi(z)\in K(z)$ is a rational function whose second iterate is not a polynomial, the set of $S$-integral points in the orbit…

We give a generalization to higher dimensions of Silverman's result on finiteness of integer points in orbits. Assuming Vojta's conjecture, we prove a sufficient condition for morphisms on P^N so that (S,D)-integral points in each orbit are…

Let $f:\mathbb{P}^1\rightarrow\mathbb{P}^1$ be a quadratic rational map defined over the rational field $\mathbb{Q}$ with nonabelian automorphism group. We prove that no such map has a $\mathbb{Q}$-rational periodic point with exact period…

We use the Weil bound of multiplicative character sums together with some recent results of N. Boston and R. Jones, to show that the critical orbit of quadratic polynomials over a finite field of $q$ elements is of length $O(q^{3/4})$,…

Fix a rational basepoint b and a rational number c. For the quadratic dynamical system f_c(x) = x^2+c, it has been shown that the number of rational points in the backward orbit of b is bounded independent of the choice of rational…

In this paper we characterise univariate rational functions over a number field $\K$ having infinitely many points in the cyclotomic closure $\K^c$ for which the orbit contains a root of unity. Our results are similar to previous results of…