2011/05/31 by Allan P. Fordy, Andrew Hone · 24 citations
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Affine transformation #Algebraic structures and combinatorial models #Cluster algebra #Combinatorics #Discrete mathematics #Integrable system #Iterated function #Mathematical analysis #Mathematics #Nonlinear Waves and Solitons #Nonlinear system #Pure mathematics #Quiver #Recurrence relation #Symplectic geometry #math-ph #math.MP #nlin.SI
paper · pdf · doi:10.3842/sigma.2011.091
published in Symmetry Integrability and Geometry Methods and Applications (National Academy of Sciences of Ukraine)
arxiv created 2011/09/22 · openalex publication_date 2011/09/22 · arxiv updated 2011/09/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.