The number of discrete Painlevé equations is infinite

Based on the geometrical description of discrete Painlevé equations, we show that one can in fact construct an infinite number of them. We present two concrete constructions, involving an arbitrary number of intermediate steps, of discrete Painlevé equations described by the affine Weyl groups A2(1) and A2(1)×A1(1). We show that the continuous limit of all these discrete systems is the same (continuous) Painlevé equation.