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

Cruz Inês; Mena-Matos Helena; Sousa-Dias Esmeralda

doi:10.3934/jgm.2020010

Article Contents

2020, Volume 12, Issue 3: 363-375. Doi: 10.3934/jgm.2020010

This issue Previous Article The method of averaging for Poisson connections on foliations and its applications Next Article Symmetry reduction of the 3-body problem in

$ \mathbb{R}^4 $

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

1.
Departamento de Matemática, Faculdade de Ciências da Universidade do Porto, R. Campo Alegre, 687, 4169-007 Porto, Portugal
2.
Center for Mathematical Analysis, Geometry and Dynamical Systems (CAMGSD), Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal

^* Corresponding author
^* Corresponding author

Received: July 2019

Early access: March 2020

Published: September 2020

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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Abstract

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 $.

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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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References

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