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A Subtle Symmetry of Lebesgue’s Measure

  • Muhammed Uludağ
  • Hakan Ayral
Article
  • 65 Downloads

Abstract

We represent the Lebesgue measure on the unit interval as a boundary measure of the Farey tree and show that this representation has a certain symmetry related to the tree automorphism induced by Dyer’s outer automorphism of the group \(\mathrm {PGL}(2,\mathbb {Z})\). Our approach gives rise to three new measures on the unit interval which are possibly of arithmetic significance.

Keywords

Farey tree Modular group Boundary Measure Lebesgue measure Stern–Brocot tree Calkin–Wilf tree 

Mathematics Subject Classification (2010)

28C10 

Notes

Acknowledgements

This research is funded by a Galatasaray University Research Grant and TÜBİTAK grant 115F412. The second named author was also funded by the TÜBİTAK grant 113R017. We are grateful to an anonymous referee for shortening the proof of the expression (18).

References

  1. 1.
    Alkauskas, G.: The Minkowski question mark function: explicit series for the dyadic period function and moments. Math. Comput. 79(269), 383–418 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alkauskas, G.: Addenda and corrigenda. Math. Comput. 80(276), 2445–2454 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bates, B., Bunder, M., Tognetti, K.: Linking the Calkin–Wilf and Stern–Brocot trees. Eur. J. Comb. 31(7), 1637–1661 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bonanno, C., Isola, S.: Orderings of the rationals and dynamical systems (2008). arXiv preprint arXiv:0805.2178
  5. 5.
    Dyer, J.L.: Automorphism sequences of integer unimodular groups. Ill. J. Math. 22(1), 1–30 (1978)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Reznick, B.: Regularity properties of the Stern enumeration of the rationals. J. Integer Seq. 11(2), 3 (2008)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Uludağ, A.M., Ayral, H.: Jimm, a fundamental involution (2015). arXiv preprint arXiv:1501.03787
  8. 8.
    Uludağ, A.M., Ayral, H.: Some deformations of Lebesgue measure on the boundary of the Farey tree (to appear)Google Scholar
  9. 9.
    Vepstas, L.: On the Minkowski Measure (2008) arXiv preprint arXiv:0810.1265

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of MathematicsGalatasaray UniversityBeşiktaşTurkey

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