  Yang-Baxter (YB) map systems (or set-theoretic analoga of entwining YB
structures) are presented. They admit zero curvature representations with
spectral parameter depended Lax triples L1, L2, L3 derived from symplectic
leaves of 2 x 2 binomial matrices equipped with the Sklyanin bracket. A unique
factorization condition of the Lax triple implies a 3-dimensional compatibility
property of these maps. In case L1 = L2 = L3 this property yields the
se--theoretic quantum Yang-Baxter equation, i.e. the YB map property. By
considering periodic 'staircase' initial value problems on quadrilateral
lattices, these maps give rise to multidimensional integrable mappings which
preserve the spectrum of the corresponding monodromy matrix.
