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Reduction of cluster iteration maps
1. | Centro de Matemática da Universidade do Porto (CMUP), 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 |
References:
[1] |
R. Abraham and J. Marsden, Foundations of Mechanics, $2^{nd}$ edition, Benjamin/Cummings Publishing Co. Inc., 1978. |
[2] |
I. Cruz and M. E. Sousa-Dias, Reduction of order of cluster-type recurrence relations, São Paulo J. Math. Sci., 6 (2012), 203-225. Available from: http://www.ime.usp.br/~spjm//articlepdf/466.pdf.
doi: 10.11606/issn.2316-9028.v6i2p203-225. |
[3] |
J. Duistermaat, Discrete Integrable Systems. QRT Maps and Elliptic Surfaces, Springer Monographs in Mathematics, Springer, New York, 2010.
doi: 10.1007/978-0-387-72923-7. |
[4] |
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. |
[5] |
A. Fordy and A. Hone, Symplectic maps from cluster algebras, SIGMA Symmetry, Integrability and Geom. Methods and Appl., 7 (2011), Paper 091, 12pp.
doi: 10.3842/sigma.2011.091. |
[6] |
A. Fordy and A. Hone, Discrete integrable systems and Poisson algebras from cluster maps, Comm. Math. Phys., 325 (2014), 527-584.
doi: 10.1007/s00220-013-1867-y. |
[7] |
A. Fordy and R. Marsh, Cluster mutation-periodic quivers and associated Laurent sequences, J. Algebraic Combin., 34 (2011), 19-66.
doi: 10.1007/s10801-010-0262-4. |
[8] |
M. Gekhtman, M. Shapiro and A. Vainshtein, Cluster Algebras and Poisson Geometry, Mathematical Surveys and Monographs, 167, AMS, Providence, RI, 2010.
doi: 10.1090/surv/167. |
[9] |
M. Gekhtman, M. Shapiro and A. Vainshtein, Cluster algebras and Weil-Petersson forms, Duke Math. J., 127 (2005), 291-311.
doi: 10.1215/S0012-7094-04-12723-X. |
[10] |
M. Gekhtman, M. Shapiro and A. Vainshtein, Cluster algebras and Poisson geometry, Mosc. Math. J., 3 (2003), 899-934. |
[11] |
A. Hone, Laurent polynomials and superintegrable maps, SIGMA Symmetry, Integrability and Geom. Methods and Appl., 3 (2007), Paper 022, 18pp.
doi: 10.3842/sigma.2007.022. |
[12] |
A. Hone and R. Inoue, Discrete Painlevé equations from Y-systems, preprint, arXiv:1405.5379. |
[13] |
A. Iatrou and J. Roberts, Integrable mappings of the plane preserving biquadratic invariant curves II, Nonlinearity, 15 (2002), 459-489.
doi: 10.1088/0951-7715/15/2/313. |
[14] |
B. Keller, Cluster algebras, quiver representations and triangulated categories, in Triangulated Categories (eds. Thorsten Holm et al.), London Math. Soc. Lecture Note Ser., 375 (2010), 76-160.
doi: 10.1017/cbo9781139107075.004. |
[15] |
P. Libermann and C-M. Marle, Symplectic Geometry and Analytical Mechanics, Mathematics and its Applications, 35, D. Reidel Publishing Co., Dordrecht, 1987.
doi: 10.1007/978-94-009-3807-6. |
[16] |
S. Sternberg, Lectures on Differential Geometry, Prentice-Hall Inc., NJ, 1964. |
[17] |
G. Quispel, J. Roberts and C. Thompson, Integrable mappings and soliton equations, Phys. Lett. A, 126 (1988), 419-421.
doi: 10.1016/0375-9601(88)90803-1. |
show all references
References:
[1] |
R. Abraham and J. Marsden, Foundations of Mechanics, $2^{nd}$ edition, Benjamin/Cummings Publishing Co. Inc., 1978. |
[2] |
I. Cruz and M. E. Sousa-Dias, Reduction of order of cluster-type recurrence relations, São Paulo J. Math. Sci., 6 (2012), 203-225. Available from: http://www.ime.usp.br/~spjm//articlepdf/466.pdf.
doi: 10.11606/issn.2316-9028.v6i2p203-225. |
[3] |
J. Duistermaat, Discrete Integrable Systems. QRT Maps and Elliptic Surfaces, Springer Monographs in Mathematics, Springer, New York, 2010.
doi: 10.1007/978-0-387-72923-7. |
[4] |
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. |
[5] |
A. Fordy and A. Hone, Symplectic maps from cluster algebras, SIGMA Symmetry, Integrability and Geom. Methods and Appl., 7 (2011), Paper 091, 12pp.
doi: 10.3842/sigma.2011.091. |
[6] |
A. Fordy and A. Hone, Discrete integrable systems and Poisson algebras from cluster maps, Comm. Math. Phys., 325 (2014), 527-584.
doi: 10.1007/s00220-013-1867-y. |
[7] |
A. Fordy and R. Marsh, Cluster mutation-periodic quivers and associated Laurent sequences, J. Algebraic Combin., 34 (2011), 19-66.
doi: 10.1007/s10801-010-0262-4. |
[8] |
M. Gekhtman, M. Shapiro and A. Vainshtein, Cluster Algebras and Poisson Geometry, Mathematical Surveys and Monographs, 167, AMS, Providence, RI, 2010.
doi: 10.1090/surv/167. |
[9] |
M. Gekhtman, M. Shapiro and A. Vainshtein, Cluster algebras and Weil-Petersson forms, Duke Math. J., 127 (2005), 291-311.
doi: 10.1215/S0012-7094-04-12723-X. |
[10] |
M. Gekhtman, M. Shapiro and A. Vainshtein, Cluster algebras and Poisson geometry, Mosc. Math. J., 3 (2003), 899-934. |
[11] |
A. Hone, Laurent polynomials and superintegrable maps, SIGMA Symmetry, Integrability and Geom. Methods and Appl., 3 (2007), Paper 022, 18pp.
doi: 10.3842/sigma.2007.022. |
[12] |
A. Hone and R. Inoue, Discrete Painlevé equations from Y-systems, preprint, arXiv:1405.5379. |
[13] |
A. Iatrou and J. Roberts, Integrable mappings of the plane preserving biquadratic invariant curves II, Nonlinearity, 15 (2002), 459-489.
doi: 10.1088/0951-7715/15/2/313. |
[14] |
B. Keller, Cluster algebras, quiver representations and triangulated categories, in Triangulated Categories (eds. Thorsten Holm et al.), London Math. Soc. Lecture Note Ser., 375 (2010), 76-160.
doi: 10.1017/cbo9781139107075.004. |
[15] |
P. Libermann and C-M. Marle, Symplectic Geometry and Analytical Mechanics, Mathematics and its Applications, 35, D. Reidel Publishing Co., Dordrecht, 1987.
doi: 10.1007/978-94-009-3807-6. |
[16] |
S. Sternberg, Lectures on Differential Geometry, Prentice-Hall Inc., NJ, 1964. |
[17] |
G. Quispel, J. Roberts and C. Thompson, Integrable mappings and soliton equations, Phys. Lett. A, 126 (1988), 419-421.
doi: 10.1016/0375-9601(88)90803-1. |
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