Symplectic Maps from Cluster Algebras

Symplectic Maps from Cluster Algebras

Blog Article

We consider nonlinear recurrences generated from the iteration of maps that arise from cluster merlin wizard costume 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 crystal beaded candle holder 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.