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On the classification of multidimensionally consistent 3D maps

Abstract

We classify multidimensionally consistent maps given by (formal or convergent) series of the following kind:

$$\begin{aligned} T_k x_{ij}=x_{ij} + \sum _{m=2}^\infty A_{ij ; \, k}^{(m)}(x_{ij},x_{ik},x_{jk}), \end{aligned}$$

where \(A_{ij;\, k}^{(m)}\) are homogeneous polynomials of degree m of their respective arguments. The result of our classification is that the only non-trivial multidimensionally consistent map in this class is given by the well-known symmetric discrete Darboux system

$$\begin{aligned} T_k x_{ij}=\frac{x_{ij}+x_{ik}x_{jk}}{\sqrt{1-x_{ik}^2}\sqrt{1-x_{jk}^2}}. \end{aligned}$$

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Acknowledgements

This research is supported by the DFG Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics.” We thank the referees for their useful remarks which helped us to improve the presentation.

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Correspondence to Yuri B. Suris.

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Petrera, M., Suris, Y.B. On the classification of multidimensionally consistent 3D maps. Lett Math Phys 107, 2013–2027 (2017). https://doi.org/10.1007/s11005-017-0976-5

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Keywords

  • Multidimensional consistency
  • Darboux system
  • Integrable three-dimensional maps

Mathematics Subject Classification

  • 37J35
  • 39A12
  • 39B12
  • 51D25