Square patterns in dynamical orbits

Abstract

Let q be an odd prime power. Let $f\in F_q\left[x\right]$ be a polynomial having degree at least 2, $a\in F_q$, and denote by $f^n$ the n-th iteration of f. Let χ be the quadratic character of $F_q$, and $O_f\left(a\right)$ the forward orbit of a under iteration by f. Suppose that the sequence ${\left(\chi \right(f^n\left(a\right)\left)\right)}_{n\ge 1}$ is periodic, and m is its period. Assuming a mild and generic condition on f, we show that, up to a constant depending on d, m can be bounded from below by $\left|O_f\right(a\left)\right|/q^{\frac{2{\log }_2\left(d\right)+1}{2{\log }_2\left(d\right)+2}}$ as q grows. 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 depending on d, we cannot have more than $q^{\frac{2{\log }_2\left(d\right)+1}{2{\log }_2\left(d\right)+2}}$ consecutive squares or non-squares in the forward orbit of a. In addition, using geometric tools from global function field theory such as abc theorem, we provide a classification of all polynomials for which our generic condition does not hold, making the results effective. Interestingly enough, our condition is purely geometrical, while our final results are completely arithmetical. As a corollary, this paper removes most of the hypothesis of (Ostafe, Shparlinski. Proceedings of the American Mathematical Society 138.8 (2010)), most notably extending the results to any degree and to non-stable polynomials.