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  September 2020 Early access  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. 

[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.

show all references

References:
[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.

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]

Rafael de la Llave, Jason D. Mireles James. Parameterization of invariant manifolds by reducibility for volume preserving and symplectic maps. Discrete and Continuous Dynamical Systems, 2012, 32 (12) : 4321-4360. doi: 10.3934/dcds.2012.32.4321

[2]

Elena Celledoni, Charalambos Evripidou, David I. McLaren, Brynjulf Owren, G. R. W. Quispel, Benjamin K. Tapley. Detecting and determining preserved measures and integrals of birational maps. Journal of Computational Dynamics, 2022  doi: 10.3934/jcd.2022014

[3]

Eric Bedford, Kyounghee Kim. Degree growth of matrix inversion: Birational maps of symmetric, cyclic matrices. Discrete and Continuous Dynamical Systems, 2008, 21 (4) : 977-1013. doi: 10.3934/dcds.2008.21.977

[4]

Fumihiko Nakamura, Michael C. Mackey. Asymptotic (statistical) periodicity in two-dimensional maps. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021227

[5]

Michael Entov, Leonid Polterovich, Daniel Rosen. Poisson brackets, quasi-states and symplectic integrators. Discrete and Continuous Dynamical Systems, 2010, 28 (4) : 1455-1468. doi: 10.3934/dcds.2010.28.1455

[6]

Andrew James Bruce, Janusz Grabowski. Symplectic $ {\mathbb Z}_2^n $-manifolds. Journal of Geometric Mechanics, 2021, 13 (3) : 285-311. doi: 10.3934/jgm.2021020

[7]

Wenxiong Chen, Congming Li. Harmonic maps on complete manifolds. Discrete and Continuous Dynamical Systems, 1999, 5 (4) : 799-804. doi: 10.3934/dcds.1999.5.799

[8]

Martin Pinsonnault. Maximal compact tori in the Hamiltonian group of 4-dimensional symplectic manifolds. Journal of Modern Dynamics, 2008, 2 (3) : 431-455. doi: 10.3934/jmd.2008.2.431

[9]

George Papadopoulos, Holger R. Dullin. Semi-global symplectic invariants of the Euler top. Journal of Geometric Mechanics, 2013, 5 (2) : 215-232. doi: 10.3934/jgm.2013.5.215

[10]

Carles Simó, Dmitry Treschev. Stability islands in the vicinity of separatrices of near-integrable symplectic maps. Discrete and Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 681-698. doi: 10.3934/dcdsb.2008.10.681

[11]

Wei Feng, Xin Lu. Global periodicity in a class of reaction-diffusion systems with time delays. Discrete and Continuous Dynamical Systems - B, 2003, 3 (1) : 69-78. doi: 10.3934/dcdsb.2003.3.69

[12]

Chi-Kwong Fok. Picard group of isotropic realizations of twisted Poisson manifolds. Journal of Geometric Mechanics, 2016, 8 (2) : 179-197. doi: 10.3934/jgm.2016003

[13]

Manuel de León, David Martín de Diego, Miguel Vaquero. A Hamilton-Jacobi theory on Poisson manifolds. Journal of Geometric Mechanics, 2014, 6 (1) : 121-140. doi: 10.3934/jgm.2014.6.121

[14]

Sze-Bi Hsu, Ming-Chia Li, Weishi Liu, Mikhail Malkin. Heteroclinic foliation, global oscillations for the Nicholson-Bailey model and delay of stability loss. Discrete and Continuous Dynamical Systems, 2003, 9 (6) : 1465-1492. doi: 10.3934/dcds.2003.9.1465

[15]

Eugen Mihailescu, Mariusz Urbański. Holomorphic maps for which the unstable manifolds depend on prehistories. Discrete and Continuous Dynamical Systems, 2003, 9 (2) : 443-450. doi: 10.3934/dcds.2003.9.443

[16]

Begoña Alarcón, Sofia B. S. D. Castro, Isabel S. Labouriau. Global dynamics for symmetric planar maps. Discrete and Continuous Dynamical Systems, 2013, 33 (6) : 2241-2251. doi: 10.3934/dcds.2013.33.2241

[17]

Daniel N. Dore, Andrew D. Hanlon. Area preserving maps on $\boldsymbol{S^2}$: A lower bound on the $\boldsymbol{C^0}$-norm using symplectic spectral invariants. Electronic Research Announcements, 2013, 20: 97-102. doi: 10.3934/era.2013.20.97

[18]

Blanca Climent-Ezquerra, Francisco Guillén-González. Global in time solution and time-periodicity for a smectic-A liquid crystal model. Communications on Pure and Applied Analysis, 2010, 9 (6) : 1473-1493. doi: 10.3934/cpaa.2010.9.1473

[19]

Toshikazu Kuniya, Mimmo Iannelli. $R_0$ and the global behavior of an age-structured SIS epidemic model with periodicity and vertical transmission. Mathematical Biosciences & Engineering, 2014, 11 (4) : 929-945. doi: 10.3934/mbe.2014.11.929

[20]

Pierre-Damien Thizy. Schrödinger-Poisson systems in $4$-dimensional closed manifolds. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 2257-2284. doi: 10.3934/dcds.2016.36.2257

2020 Impact Factor: 0.857

Metrics

  • PDF downloads (223)
  • HTML views (350)
  • Cited by (0)

[Back to Top]