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Square patterns in dynamical orbits

Vefa Goksel, Giacomo Micheli

Abstract

Let $q$ be an odd prime power. Let $f\in \mathbb{F}_q[x]$ be a polynomial having degree at least $2$, $a\in \mathbb{F}_q$, and denote by $f^n$ the $n$-th iteration of $f$. Let $χ$ be the quadratic character of $\mathbb{F}_q$, and $\mathcal{O}_f(a)$ the forward orbit of $a$ under iteration by $f$. Suppose that the sequence $(χ(f^n(a)))_{n\geq 1}$ is periodic, and $m$ is its period. Assuming a mild and generic condition on $f$, we show that, up to a constant, $m$ can be bounded from below by $|\mathcal{O}_f(a)|/q^\frac{2\log_{2}(d)+1}{2\log_2(d)+2}$. More informally, we prove that the period of the appearance of squares in an orbit of an element provides an upper bound for the size of the orbit itself. Using a similar method, we can also prove that, up to a constant, we cannot have more than $q^\frac{2\log_2(d)+1}{2\log_2(d)+2}$ consecutive squares or non-squares in the forward orbit of $a$. In addition, we provide a classification of all polynomials for which our generic condition does not hold.

Square patterns in dynamical orbits

Abstract

Let be an odd prime power. Let be a polynomial having degree at least , , and denote by the -th iteration of . Let be the quadratic character of , and the forward orbit of under iteration by . Suppose that the sequence is periodic, and is its period. Assuming a mild and generic condition on , we show that, up to a constant, can be bounded from below by . More informally, we prove that the period of the appearance of squares in an orbit of an element provides an upper bound for the size of the orbit itself. Using a similar method, we can also prove that, up to a constant, we cannot have more than consecutive squares or non-squares in the forward orbit of . In addition, we provide a classification of all polynomials for which our generic condition does not hold.
Paper Structure (4 sections, 17 theorems, 65 equations)

This paper contains 4 sections, 17 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Dynamically ordinary polynomials
  3. dynamically $2$-ordinary polynomials
  4. Square patterns in orbits

Key Result

Theorem 1.1

Let $q=p^k$ for some odd prime $p$ and $k\geq 1$, and let $d:=\deg(f)\geq 2$. Suppose that $f\in \mathbb{F}_q[x]$ is not in one of the following forms. Let $a\in \mathbb{F}_q$. Suppose that the sequence $(\chi(f^n(a)))_{n\geq 0}$ is periodic, and let $m:=m_a$ be its period. Then and the implied constant is only dependent on $d$.

Theorems & Definitions (39)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Definition 1.4
  • Theorem 1.5
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • ...and 29 more