Thursday, July 26, 2012

1207.6072 (Allan Fordy et al.)

Discrete integrable systems and Poisson algebras from cluster maps    [PDF]

Allan Fordy, Andrew Hone
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 introduced by Bellon and Viallet, 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: (i) periodic maps; (ii) maps arising from affine $A$-type Dynkin quivers, whose iterates satisfy linear recurrence relations; (iii) deformations of (ii), whose iterates also satisfy linear recurrence relations; and (iv) Somos-type maps. The families (ii), (iii) and (iv) are examined in detail, with many explicit examples given, and we show how these particular families lead to discrete dynamics that is integrable in the Liouville-Arnold sense. The connection with Y-systems and T-systems, in the general setting due to Nakinishi, is also described, along with an example where a discrete Painlev\'e equation arises.
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