September  2020, 12(3): 363-375. doi: 10.3934/jgm.2020010

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

Received  July 2019 Published  March 2020

Fund Project: 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

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 $.

Citation: Inês Cruz, Helena Mena-Matos, Esmeralda Sousa-Dias. The group of symplectic birational maps of the plane and the dynamics of a family of 4D maps. Journal of Geometric Mechanics, 2020, 12 (3) : 363-375. doi: 10.3934/jgm.2020010
References:
[1]

J. Blanc, Symplectic birational transformations of the plane, Osaka J. Math., 50 (2013), 573-590.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

I. Cruz, H. Mena-Matos and M. E. Sousa-Dias, Dynamics and periodicity in a family of cluster maps, preprint, arXiv: 1511.07291. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

R. J. Marsh, Lecture Notes on Cluster Algebras, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2013.  Google Scholar

show all references

References:
[1]

J. Blanc, Symplectic birational transformations of the plane, Osaka J. Math., 50 (2013), 573-590.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

I. Cruz, H. Mena-Matos and M. E. Sousa-Dias, Dynamics and periodicity in a family of cluster maps, preprint, arXiv: 1511.07291. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

R. J. Marsh, Lecture Notes on Cluster Algebras, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2013.  Google Scholar

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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