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 $.

中文翻译：

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平面的辛双平映射组和4D映射族的动力学

我们考虑一个维数为4的双分对映图族$ \ varphi_k $，它是在周期2的突变周期颤动中的簇代数的背景下产生的。我们使用Poisson几何工具，即$映射$ \ varphi_k $和它们的第四个迭代$ \ varphi ^ {（4）} _ k $的约束的性质到适当的泊松流形$（\ mathbb {R} ^ 4_ +，P）$的辛叶。这些受限制的映射显示为属于平面的一组辛双平映射，其与半直接乘积$ SL（2，\ mathbb {Z}）\ ltimes \ mathbb {R} ^ 2 $同构。对这些受限映射的研究得出结论，对于$ \ varphi_k $，存在三种不同类型的动力学行为，其特征在于参数值$ k = 1 $，$ k = 2 $和$ k \ geq 3 $。