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

Abstract

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