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Finite Gauss Transformations

  • Mathematical Gems and Curiosities
  • Edited by Sophie Morier-Genoud and Valentin Ovsienko
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Figure 1.
Figure 2.
Figure 3.

Notes

  1. The transformation \({\textsf {R}}\) is called a hyperbolic reflection in the ideal line through the endpoints of the arc \(A_0\), and the return maps \({\textsf {T}}_j\) are called hyperbolic translations. The interested reader may refer, for instance, to [3, Chapter 19] for a comprehensive survey on hyperbolic geometry.

  2. The transformation (1) is similar to the Romik transform [8] with the only difference that our transformation is orientation-reversing on each interval of the partition, while the Romik transform is orientation-reversing on only the middle interval.

References

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  4. O. I. Bogojavlenskii. Qualitative theory of homogeneous cosmological models II. Trudy Sem. Petrovsk. 2 (1976), 67–112.

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  5. S. Chaubey, E. Fuchs, R. Hines, and K. E. Stange. The dynamics of super-Apollonian continued fractions. Trans. Amer. Math. Soc. 372:4 (2019), 2287–2334.

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Acknowledgments

The authors are deeply indebted to Yakov Grigorevich Sinai and Oleg Igorevich Bogoyavlenskij for providing the reference [4], where one could probably find the proof of our theorem for the case \(n=3\). The authors are also grateful to Yuliy Baryshnikov and Serge Tabachnikov for inspiring discussions.

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  1. University of Texas at Dallas, 800 W. Campbell Rd., Richardson, TX, 75080, USA

    Maxim Arnold & Anatoly Eydelzon

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  1. Maxim Arnold
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  2. Anatoly Eydelzon
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Correspondence to Maxim Arnold.

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Arnold, M., Eydelzon, A. Finite Gauss Transformations. Math Intelligencer 45, 159–164 (2023). https://doi.org/10.1007/s00283-022-10198-7

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