Arithmetic dynamics of a discrete Painlev\'e equation

We consider the orbits of a discrete Painlev\'e equation over finite fields and show that the number of points in such orbits satisfy the Hasse bound. The orbits turn out to lie on algebraic curves, whose defining polynomials are given explicitly. Moreover, these curves are shown to have genus less than or equal to one, which contrasts sharply with the case of discrete Painlev\'e equations over $\mathbb{C}$, whose generic solutions are believed to be more transcendental than elliptic functions.