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

Abstract

<p style='text-indent:20px;'>We consider a family of birational maps <inline-formula><tex-math id="M1">\begin{document}$ \varphi_k $\end{document}</tex-math></inline-formula> in dimension 4, arising in the context of cluster algebras from a mutation-periodic quiver of period 2. We approach the dynamics of the family <inline-formula><tex-math id="M2">\begin{document}$ \varphi_k $\end{document}</tex-math></inline-formula> using Poisson geometry tools, namely the properties of the restrictions of the maps <inline-formula><tex-math id="M3">\begin{document}$ \varphi_k $\end{document}</tex-math></inline-formula> and their fourth iterate <inline-formula><tex-math id="M4">\begin{document}$ \varphi^{(4)}_k $\end{document}</tex-math></inline-formula> to the symplectic leaves of an appropriate Poisson manifold <inline-formula><tex-math id="M5">\begin{document}$ (\mathbb{R}^4_+, P) $\end{document}</tex-math></inline-formula>. These restricted maps are shown to belong to a group of symplectic birational maps of the plane which is isomorphic to the semidirect product <inline-formula><tex-math id="M6">\begin{document}$ SL(2, \mathbb{Z})\ltimes\mathbb{R}^2 $\end{document}</tex-math></inline-formula>. The study of these restricted maps leads to the conclusion that there are three different types of dynamical behaviour for <inline-formula><tex-math id="M7">\begin{document}$ \varphi_k $\end{document}</tex-math></inline-formula> characterized by the parameter values <inline-formula><tex-math id="M8">\begin{document}$ k = 1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M9">\begin{document}$ k = 2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M10">\begin{document}$ k\geq 3 $\end{document}</tex-math></inline-formula>.