Symplectic Maps from Cluster Algebras

Abstract

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.

References in corpus (3)

• Algebraic entropy
• Discrete integrable systems and Poisson algebras from cluster maps
• Mutation-Periodic Quivers, Integrable Maps and Associated Poisson Algebras

Cited by in corpus (9)

• Discrete integrable systems and Poisson algebras from cluster maps
• On the algebraic structure of rational discrete dynamical systems
• A compendium on the cluster algebra and quiver package in sage
• Hankel determinant solutions to several discrete integrable systems and the Laurent Property
• Integrability of reductions of the discrete KdV and potential KdV equations
• Dynamics of the birational maps arising from $F_0$ and $d{P}_3$ quivers
• From integrable equations to Laurent recurrences
• Poisson structures for difference equations
• Dynamics and periodicity in a family of cluster maps