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Calculation of algebraic entropies of d-\(P_{IV}\) and d-\(P_V\). (English) Zbl 1486.37033

The authors compute the algebraic entropy of the discrete Painlevé IV and discrete Painlevé V equations, that is, the asymptotic exponential rate of the degree of \(n\) iterations of the equations, and show that it is zero. They in fact give exact (polynomial) expressions for such degrees.
Their proof is based on a method of R. G. Halburd [Proc. R. Soc. Lond., A, Math. Phys. Eng. Sci. 473, No. 2201, Article ID 20160831, 13 p. (2017; Zbl 1404.39020)], that relies on the study of singularities of the equation. It goes through the proof of the singularity confinement property, that is, if some initial data are singular, they can be deformed to give coherent values to iterations of the data, thus escaping the singularity. Other types of singularity are discussed: unconfined, i.e., singularities always reappear after a number of steps; cyclic, i.e., they appear cyclically; anti-confined, i.e., they appear in both the forward and backwards dynamics.
Then the authors consider an example that contradicts the idea that singularity confinement and zero algebraic entropy are both indicators of integrability. It is a modification of discrete Painlevé IV equation: \[(x_{n+1} + x_n) (x_n + x_{n-1}) = \frac{1}{x_n^m}.\] This difference equation has singularity confinement and positive algebraic entropy. In a second example, they show that cyclic and anti-confined singularity can also play a crucial role when computing the algebraic entropy.

MSC:

37J70 Completely integrable discrete dynamical systems
39A36 Integrable difference and lattice equations; integrability tests
30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
39A12 Discrete version of topics in analysis

Citations:

Zbl 1404.39020
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References:

[1] Halburd, R.; Korhonen, R., Comput. Methods Funct. Theory, 19, 299 (2019) · Zbl 1423.30024 · doi:10.1007/s40315-019-00271-2
[2] Hietarinta, J.; Viallet, C., Phys. Rev. Lett., 81, 325 (1998) · doi:10.1103/physrevlett.81.325
[3] van der Kamp, P. H., J. Differ. Equations Appl., 18, 447 (2012) · Zbl 1238.39003 · doi:10.1080/10236198.2010.510137
[4] Halburd, R. G., Proc. R. Soc. A, 473, 20160831 (2017) · Zbl 1404.39020 · doi:10.1098/rspa.2016.0831
[5] Mase, T.; Willox, R.; Ramani, A.; Grammaticos, B., J. Phys. A: Math. Theor., 51, 265201 (2018) · Zbl 1405.37063 · doi:10.1088/1751-8121/aac578
[6] Ramani, A.; Grammaticos, B.; Willox, R.; Mase, T.; Satsuma, J., J. Integr. Syst., 3, xyy006 (2018) · Zbl 1397.37062 · doi:10.1093/integr/xyy006
[7] Ramani, A.; Grammaticos, B.; Willox, R.; Mase, T., J. Phys. A: Math. Theor., 50, 185203 (2017) · Zbl 1390.81106 · doi:10.1088/1751-8121/aa66d7
[8] Ramani, A.; Grammaticos, B.; Hietarinta, J., Phys. Rev. Lett., 67, 1829 (1991) · Zbl 1050.39500 · doi:10.1103/physrevlett.67.1829
[9] Ohta, Y.; Tamizhmani, K. M.; Grammaticos, B.; Ramani, A., Phys. Lett. A, 262, 152 (1999) · Zbl 0939.37041 · doi:10.1016/s0375-9601(99)00670-2
[10] Tomoyuki, T., J. Phys. A: Math. Gen., 34, 10533 (2001) · Zbl 0999.37028 · doi:10.1088/0305-4470/34/48/317
[11] Grammaticos, B.; Ramani, A.; Lafortune, S., Physica A, 253, 260 (1998) · Zbl 0910.58017 · doi:10.1016/s0378-4371(97)00675-4
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