A new family of four-dimensional symplectic and integrable mappings

We investigate the generalisations of the Quispel, Roberts and Thompson (QRT) family of mappings in the plane leaving a rational quadratic expression invariant to the case of four variables. We assume invariance of the rational expression under a cyclic permutation of variables and we impose a symplectic structure with Poisson brackets of the Weyl type. All mappings satisfying these conditions are shown to be integrable either as four-dimensional mappings with two explicit integrals which are in involution with respect to the symplectic structure and which can also be defined from the periodic reductions of the double-discrete versions of the modified Korteweg-de Vries $(\delta\delta MkdV)$ and sine-Gordon $(\delta\delta sG)$ equations or by reduction to two-dimensional mappings with one integral of the symmetric QRT family.