Abstract

For discrete equations on a number field, the rate of growth of the heights of iterates is a good detector of integrability. In the case of a rational number, the height is just the maximum of the absolute value of the denominator and numerator. A solution is called admissible if its height grows much faster than the heights of the coefficients in the equation. For certain classes of equations it is shown that the existence of a single slow-growing admissible solution is enough to guarantee that the equation is a discrete Painleve equation. Inadmissible solutions are also explored. These solutions correspond to pre-periodic orbits for classical (autonomous) dynamical systems. The classical theory is extended to better understand these solutions in the non-autonomous setting.