  We consider nonlinear recurrences generated from the iteration of maps that
arise from cluster algebras. More precisely, starting from a skew-symmetric
integer matrix, or its corresponding quiver, one can define a set of mutation
operations, as well as a set of associated cluster mutations that are applied
to a set of affine coordinates (the cluster variables). Fordy and Marsh
recently provided a complete classification of all such quivers that have a
certain periodicity property under sequences of mutations. This periodicity
implies that a suitable sequence of cluster mutations is precisely equivalent
to iteration of a nonlinear recurrence relation. Here we explain briefly how to
introduce a symplectic structure in this setting, which is preserved by a
corresponding birational map (possibly on a space of lower dimension). We give
examples of both integrable and non-integrable maps that arise from this
construction. We use algebraic entropy as an approach to classifying integrable
cases. The degrees of the iterates satisfy a tropical version of the map.
