Phase Transition of Logarithmic Capacity for the Uniform Gδ-Sets

Abstract

We consider a family of dense Gδ subsets of [0, 1], defined as intersections of unions of small uniformly distributed intervals, and study their logarithmic capacity. Changing the speed at which the lengths of generating intervals decrease, we observe a sharp phase transition from full to zero capacity. Such a Gδ set can be considered as a toy model for the set of exceptional energies in the parametric version of the Furstenberg theorem on random matrix products. Our re-distribution construction can be considered as a generalization of a method applied by Ursell in his construction of a counter-example to a conjecture by Nevanlinna. Also, we propose a simple Cauchy-Schwartz inequality-based proof of related theorems by Lindeberg and by Erdös and Gillis.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Carleson, L.: On the connection between Hausdorff measures and capacity. Arkiv för Matematik, 3 5, 403–406 (1958)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Deny, J.: Sur les infinis d’un potentiel. C. R. Acad. Sci. Paris 224, 524–525 (1947)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Erdös, P., Gillis, J.: Note on the transfinite diameter. J. London Math. Soc. 12(3), 185–192 (1937)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Gorodetski, A., Kleptsyn, V.: Parametric Furstenberg Theorem on Random Products of \(SL(2,\mathbb {R})\) matrices, preprint. arXiv:1809.00416

  5. 5.

    Helms, L.: Potential Theory. Springer (2009)

  6. 6.

    Lindeberg, J.W.: Sur l’existence des fonctions d’une variable complexe et des fonctions harmoniques bornées. Ann. Acad. Scient. Fenn. 11, 6 (1918)

    MATH  Google Scholar 

  7. 7.

    Myrberg, P.J.: Über die Existenz Der Greenschen Funktionen auf Einer Gegebenen Riemannschen Fläche. Acta Math. 61, 39–79 (1933)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Nevanlinna, R.: Über die Kapazität der Cantorschen Punktmengen. Monatshefte für Mathematik und Physik 43, 435–447 (1936)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Ransford, T.: Potential Theory in the Complex Plane. Cambridge University Press (1995)

  10. 10.

    Simon, B.: Equilibrium measures and capacities in spectral theory. Inverse Problems Imaging 1(4), 713–772 (2007)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Ursell, H.: Note on the transfinite diameter. J. London Math. Soc. 13(1), 34–37 (1938)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgments

Victor Kleptsyn was partially supported by the project ANR Gromeov (ANR-19-CE40-0007), as well as by the Laboratory of Dynamical Systems and Applications NRU HSE, of the Ministry of science and higher education of the RF grant ag. No 075-15-2019-1931. Fernando Quintino was supported by NSF grants DMS-1855541 and DMS-1700143.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Fernando Quintino.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Kleptsyn, V., Quintino, F. Phase Transition of Logarithmic Capacity for the Uniform Gδ-Sets. Potential Anal (2021). https://doi.org/10.1007/s11118-020-09896-8

Download citation

Keywords

  • Logarithmic capacity
  • Phase transition
  • Parametric Furstenberg theorem

Mathematics Subject Classification (2010)

  • Primary: 31A15 · 31C15
  • Secondary: 28A12