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Letters in Mathematical Physics

, Volume 107, Issue 11, pp 2013–2027 | Cite as

On the classification of multidimensionally consistent 3D maps

  • Matteo Petrera
  • Yuri B. Suris
Article
  • 72 Downloads

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

Keywords

Multidimensional consistency Darboux system Integrable three-dimensional maps 

Mathematics Subject Classification

37J35 39A12 39B12 51D25 

Notes

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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Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.Institut für Mathematik, MA 7-1Technische Universität BerlinBerlinGermany

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