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The group of symplectic birational maps of the plane and the dynamics of a family of 4D maps

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The work of the first and second authors is partially funded by FCT (Portugal) under the project PEst-C/MAT/UI0144/2013. The third author is partially funded by FCT (Portugal) under the projects UID/MAT/04459/2013 and PTDC/MAT-PUR/29447/2017

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  • We consider a family of birational maps $ \varphi_k $ in dimension 4, arising in the context of cluster algebras from a mutation-periodic quiver of period 2. We approach the dynamics of the family $ \varphi_k $ using Poisson geometry tools, namely the properties of the restrictions of the maps $ \varphi_k $ and their fourth iterate $ \varphi^{(4)}_k $ to the symplectic leaves of an appropriate Poisson manifold $ (\mathbb{R}^4_+, P) $. These restricted maps are shown to belong to a group of symplectic birational maps of the plane which is isomorphic to the semidirect product $ SL(2, \mathbb{Z})\ltimes\mathbb{R}^2 $. The study of these restricted maps leads to the conclusion that there are three different types of dynamical behaviour for $ \varphi_k $ characterized by the parameter values $ k = 1 $, $ k = 2 $ and $ k\geq 3 $.

    Mathematics Subject Classification: Primary: 53D17, 37J10; Secondary: 57R30, 37J15, 39A20, 13F60.

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  • Figure 1.  Quiver associated to the family of maps $ \varphi_k $. The label on the arrows indicates the number of arrows between the nodes

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