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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

• 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.

 Citation:

• 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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