Square patterns in dynamical orbits

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 $\chi$ 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 $(\chi (f^n(a)){)}_{n\ge 1}$ 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.

17 pages