A systematic approach to reductions of type-Q ABS equations

We present a class of reductions of Möbius type for the lattice equations known as Q1, Q2, and Q3 from the ABS list. The deautonomized form of one particular reduction of Q3 is shown to exist on the A₁ ⁽¹⁾surface which belongs to the multiplicative type of rational surfaces in Sakais classification of Painlevé systems. Using the growth of degrees of iterates, all other mappings that result from the class of reductions considered here are shown to be linearizable. Any possible linearizations are calculated explicitly by constructing a birational transformation defined by invariant curves in the blown up space of initial values for each reduction.