Square patterns in dynamical orbits

Let q be an odd prime power. Let f∈ Fq[x] be a polynomial having degree at least $2$, a∈ Fq, and denote by fⁿ the n-th iteration of f. Let χ be the quadratic character of Fq, and Of(a) the forward orbit of a under iteration by f. Suppose that the sequence (χ(fⁿ(a)))n≥ 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)|/q^2log₂(d)+12log₂(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 q²log₂(d)+1/2log₂(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.