Skip to main content
SpringerLink
Log in

On Möbius orthogonality for interval maps of zero entropy and orientation-preserving circle homeomorphisms

  • Published:
Arkiv för Matematik

Abstract

We will prove Sarnak’s conjecture on Möbius disjointness for continuous interval maps of zero entropy and also for orientation-preserving circle homeomorphisms by reducing these result to a well-known theorem of Davenport from 1937.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Blokh, A. M., Limit behavior of one-dimensional dynamic systems, Uspekhi Mat. Nauk 37 (1982), 137–138 (Russian). English transl.: Russian Math. Surveys 37 (1982), 157–158.

    MathSciNet  Google Scholar 

  2. Bourgain, J., Sarnak, P. and Ziegler, T., Disjointness of Moebius from horocycle flows, in From Fourier Analysis and Number Theory to Radon Transforms and Geometry, pp. 67–83, Springer, New York, 2013.

    Chapter  Google Scholar 

  3. Davenport, H., On some infinite series involving arithmetical functions. II, Quart. J. Math. Oxford Ser. 8 (1937), 313–320.

    Article  Google Scholar 

  4. Green, B. and Tao, T., The Moebius function is strongly orthogonal to nilsequences, Ann. of Math. 175 (2012), 541–566.

    Article  MathSciNet  MATH  Google Scholar 

  5. Katok, A. and Haselblatt, B., Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge, 1995.

    Book  MATH  Google Scholar 

  6. Ruette, S., Chaos for continuous interval maps, Preprint, 2003. http://www.math.u-psud.fr/~ruette/articles/chaos-int.pdf.

  7. Sarnak, P., Three lectures on the Mobius function randomness and dynamics, Preprint, 2011. http://www.math.ias.edu/files/wam/2011/PSMobius.pdf.

  8. Smítal, J., Chaotic functions with zero topological entropy, Trans. Amer. Math. Soc. 297 (1986), 269–282.

    Article  MathSciNet  Google Scholar 

  9. Vinogradov, I. M., Some theorems concerning the theory of primes, Mat. Sb. 2 (1937), 179–195.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, KTH Royal Institute of Technology, SE-100 44, Stockholm, Sweden

    Davit Karagulyan

Authors
  1. Davit Karagulyan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Davit Karagulyan.

Additional information

The author would like to express his gratitude to Ana Rodrigues for proposing the problem and for useful comments on the manuscript, and to Michael Benedicks for his guidance and many valuable suggestions. I also want to thank Lennart Carleson for his suggestion to consider circle homeomorphisms that are only semi-conjugate to rotations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Karagulyan, D. On Möbius orthogonality for interval maps of zero entropy and orientation-preserving circle homeomorphisms. Ark Mat 53, 317–327 (2015). https://doi.org/10.1007/s11512-014-0208-5

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11512-014-0208-5

Keywords

Search

Navigation