Cluster maps are birational maps arising from mutation-periodic quivers. These quivers are associated to (mutation-periodic) cluster algebras, introduced in 2002 by Fomin and Zelevinsky.

The mutation-periodic property guarantees the existence of a pre-symplectic structure which is invariant under the cluster map, as proved in 2011 for 1- periodic quivers (A. Fordy and A. Hone) and in 2014 for arbitrary periodic quivers (I. Cruz and E. Sousa-Dias). The dynamics of cluster maps is usually studied by using this invariant pre-symplectic form, which allows for the reduction to lower dimensions as long as the matrix defining the quiver is singular.

In this seminar I will review the pre-symplectic approach to reduction and introduce another approach to reduction based on the use of Poisson structures, enhancing the role of the null foliation and of the symplectic foliation canonically defined by these structures. The main purpose of the talk is to explain how to combine the two distinct approaches to gain a better insight into the dynamics of the cluster map. Time allowing, I will discuss two examples: the Somos-5 and an instance of the Somos-7 cluster maps.