Square patterns in dynamical orbits

Let q be an odd prime power. Let f Fqx be a polynomial having degree at least 2, a Fq, and denote by fⁿ the n-th iteration of f. Let be the quadratic character of Fq, and Of (a) the forward orbit of a under iteration by f. Suppose that the sequence ( (fⁿ (a) ) ) ₍ 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 |Of (a) |/q²₂ (₃) +₁2₂ (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 q²₂ (d) +12₂ (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.