Integrable boundary conditions for quad equations, open boundary reductions and integrable mappings

In the context of integrable difference equations on quad-graphs, we introduce the method of open boundary reductions, as an alternative to the well-known periodic reductions, for constructing discrete integrable mappings and their invariants. The mappings are obtained from well-posed initial value problems for quad and boundary equations restricted to strips on Z2-lattices. The invariants are constructed using Sklyanin’s double-row monodromy matrix. To establish its properties, we use the discrete zero curvature condition and boundary zero curvature condition, showing how the latter derives from the boundary consistency condition. We focus on the Adler–Bobenko–Suris classification and associated integrable boundary equations. Examples are given for the H1 and Q1(⁠δ=0⁠) equations, leading to novel maps of the plane.