Cluster mutation-periodic quivers and associated Laurent sequences

Understand

We consider quivers/skew-symmetric matrices under the action of mutation (in the cluster algebra sense).

• We classify those which are isomorphic to their own mutation via a cycle permuting all the vertices, and give families of quivers which have higher periodicity.
• The periodicity means that sequences given by recurrence relations arise in a natural way from the associated cluster algebras.
• We present a number of interesting new families of non-linear recurrences, necessarily with the Laurent property, of both the real line and the plane, containing integrable maps as special cases.