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. In particular, we show
that some of these recurrences can be linearised and, with certain initial
conditions, give integer sequences which contain all solutions of some
particular Pell equations. We extend our construction to include recurrences
with parameters, giving an explanation of some observations made by Gale.
Finally, we point out a connection between quivers which arise in our
classification and those arising in the context of quiver gauge theories.Comment: The final publication is available at www.springerlink.com. 42 pages,
35 figure
We consider nonlinear recurrences generated from cluster mutations applied to quivers that have the property of being cluster mutation-periodic with period 1. Such quivers were completely classified by Fordy and Marsh, who characterised them in terms of the skew-symmetric matrix that defines the quiver. The associated nonlinear recurrences are equivalent to birational maps, and we explain how these maps can be endowed with an invariant Poisson bracket and/or presymplectic structure.Upon applying the algebraic entropy test, we are led to a series of conjectures which imply that the entropy of the cluster maps can be determined from their tropical analogues, which leads to a sharp classification result. Only four special families of these maps should have zero entropy. These families are examined in detail, with many explicit examples given, and we show how they lead to discrete dynamics that is integrable in the Liouville-Arnold sense.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.