Advanced Search
Article Contents
Article Contents

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

  • * Corresponding author

    * Corresponding author

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

Abstract Full Text(HTML) Figure(1) Related Papers Cited by
  • 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.


    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure 1.  Quiver associated to the family of maps $ \varphi_k $. The label on the arrows indicates the number of arrows between the nodes

  • [1] J. Blanc, Symplectic birational transformations of the plane, Osaka J. Math., 50 (2013), 573-590. 
    [2] I. Cruz and M. E. Sousa-Dias, Reduction of cluster iteration maps, Journal of Geometric Mechanics, 6 (2014), 297-318.  doi: 10.3934/jgm.2014.6.297.
    [3] I. CruzH. Mena-Matos and M. E. Sousa-Dias, Dynamics of the birational maps arising from $F_0$ and $dP_3$ quivers, Journal of Mathematical Analysis and Applications, 431 (2015), 903-918.  doi: 10.1016/j.jmaa.2015.06.017.
    [4] I. Cruz, H. Mena-Matos and M. E. Sousa-Dias, Dynamics and periodicity in a family of cluster maps, preprint, arXiv: 1511.07291.
    [5] I. CruzH. Mena-Matos and M. E. Sousa-Dias, Multiple reductions, foliations and the dynamics of cluster maps, Regular and Chaotic Dynamics, 23 (2018), 102-119.  doi: 10.1134/S1560354718010082.
    [6] S. Fomin and A. Zelevinsky, Cluster algebras. I. Foundations, J. Amer. Math. Soc., 15 (2002), 497-529.  doi: 10.1090/S0894-0347-01-00385-X.
    [7] A. P. Fordy and A. Hone, Discrete integrable systems and Poisson algebras from cluster maps, Commun. Math. Phys., 325 (2014), 527-584.  doi: 10.1007/s00220-013-1867-y.
    [8] A. P. Fordy and A. Hone, Symplectic maps from cluster algebras, Symmetry, Integrability and Geometry: Methods and Applications, 7, (2011), 12 pp. doi: 10.3842/SIGMA.2011.091.
    [9] A. P. Fordy and R. J. Marsh, Cluster mutation-periodic quivers and associated Laurent sequences, J. Algebraic Combin., 34 (2011), 19-66.  doi: 10.1007/s10801-010-0262-4.
    [10] M. Gekhtman, M. Shapiro and A. Vainshtein, Cluster Algebras and Poisson Geometry, Mathematical Surveys and Monographs, 167. American Mathematical Society, Providence, RI, 2010. doi: 10.1090/surv/167.
    [11] R. J. Marsh, Lecture Notes on Cluster Algebras, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2013.
  • 加载中



Article Metrics

HTML views(533) PDF downloads(243) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint