Binning phenomenon for reduced orbit lengths of qP over finite fields

Determine whether, for the discrete Painlevé mapping (x,y,t) ↦ (t/(x−s^{-1}y), s x/y, s t) over a finite field F_q with s ∈ F_q^* of multiplicative order r, the reduced orbit length #(γ)/r always lies in one of the M_q disjoint bins B_m^{(q)}, where M_q = ⌈(1/4)(√q + 1/(√q − 2))⌉ and B_m^{(q)} = [q+1−2√q·m, q+1+2√q·m] for 1 ≤ m ≤ M_q−1 and B_{M_q}^{(q)} = [1, q+1+2√q·M_q].

Background

Beyond the Hasse-type upper bound, the computational data reveal a striking refinement: for fixed q, the reduced orbit lengths cluster into specific intervals (“bins”) centered around q+1 with widths proportional to √q, reminiscent of distributions of point counts on elliptic curves.

The authors formalize this observation by defining a finite number M_q of bins and conjecturing that every reduced orbit length falls into one of these bins, providing a structured constraint on the spectrum of orbit sizes.