A new family of four-dimensional symplectic and integrable mappings

We investigate the generalisations of the Quispel, Roberts and Thompson (QR T) family of mappings in the plane leaving a rational quadratic expression invariant to the case of four variables. We assume invariance of the ration al expression under a cyclic permutation of variables and we impose a sympl ectic structure with Poisson brackets of the Weyl type. All mappings satisf ying 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 inferred from the periodi c reductions of the double-discrete versions of the modified Korteweg-deVri es (Delta Delta MKdV) and sine-Gordon (Delta Delta sG) equations or by redu ction to two-dimensional mappings with one integral of the symmetric QRT fa mily. (C) 2001 Elsevier Science B.V. All rights reserved.