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Presymplectic and Poisson reduction of cluster maps are described in terms ofthe "canonical" foliations of presymplectic and Poisson manifolds. Thisapproach to reduction leads to a geometric description, in terms of foliations,of the dynamics of the original (not reduced) map. The case where multiplereductions exist (presymplectic/Poisson or presymplectic/Poisson/Poisson) isfurther explored and examples illustrating several features of this approachare presented, including a nontrivial one in dimension seven which iscomprehensively treated.

Multiple reductions, foliations and the dynamics of cluster maps