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Divergence points of self-conformal measures

  • Pei Wang
  • Yong JiEmail author
  • Ercai Chen
  • Yaqing Zhang
Article
  • 9 Downloads

Abstract

In this article, let \(\mu \) be a self-conformal measure, we discuss the dimensions of divergence points of self-conformal measures with the open set condition. Our main result is that the set \(\{x\in \mathrm{{{\,\mathrm{supp}\,}}}\mu : A(\frac{\log \mu (B(x,r))}{\log r})=I\}\) is not Taylor fractal and the set \(\{x\in \mathrm{{{\,\mathrm{supp}\,}}}\mu : A(\frac{\log \mu (B(x,r))}{\log r})\subseteq I\}\) is Taylor fractal.

Keywords

Self-conformal measure Divergence points Moran structrue Dimension Open set condition 

Mathematics Subject Classification

54H20 

Notes

Acknowledgements

The third author was supported by NNSF of China (11671208 and 11271191).

References

  1. 1.
    Arbeiter, M., Patzschke, N.: Random self-similar multifractals. Math. Nach. 181, 5–42 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Beak, I., Olsen, L., Snigireva, N.: Divergence points of self-similar measures and packing dimension. Adv. Math. 214, 267–287 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Barreira, L., Schmeling, J.: Sets of ’non-typical’ points have full topological entropy and full Hausdorff dimension. Isr. J. Math. 116, 29–70 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chen, E., Xiong, J.: The pointwise dimension of self-similar measures. Chin. Sci. Bull. 44, 2136–2140 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Dai, Meifeng, Li, Wenwen: The mixed \(L^q\)-spectra of self-conformal measures satisfying the weak separation condition. J. Math. Anal. Appl. 382, 140–147 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Falconer, K.: The Geometry of Fractal Sets. Cambridge University Press, Cambridge (1985)CrossRefzbMATHGoogle Scholar
  7. 7.
    Feng, D., Lau, K.: Multifractal formalism for self-similar measures with weak separation condition. J. Math. Pures Appl. 92, 407–428 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Feng, D., Lau, K., Wu, J.: Ergodic limits on the conformal repellars. Adv. Math. 169, 58–91 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Feng, D., Hu, H.: Dimension theory of iterated function systems. Commun. Pure Appl. Math. 62, 1435–1500 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hentschel, H., Procaccia, I.: The infinite number of generalized dimensions of fractals and strange attractors. Phys. D 8, 435–444 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hutchinson, J.: Fractals and self-similarity. Indiana Univ. Math. J. 30, 713–747 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Li, J., Wu, M., Xiong, Y.: Hausdorff dimensions of divergence points of self-similar measures with the open set conditon. Nonlinearity 25, 93–105 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Mihailescu, E., Urbański, M.: Random countable iterated function systems with overlaps and applications. Adv. Math. 298, 726–758 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Olsen, L., Winter, S.: Normal and non-normal points of self-similar sets and divergence points of self-similar measures. J. Lond. Math. Soc. (2) 67, 103–122 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Olsen, L.: A multifractal formalism. Adv. Math. 116, 82–196 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Patzschke, N.: Self-conformal multifractal measure. Adv. Appl. Math. 19, 486–513 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Pesin, Y., Weiss, H.: A multifractal analysis of equilibrium measures for conformal expanding maps and Moran-like geometric constructions. J. Stat. Phys. 86, 233–275 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Strichartz, R.: Self-similar measures and their Fourier transforms. Indiana Univ. Math. J. 39, 797–817 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Xiao, J., Wu, M., Gao, F.: Divergence points of self-similar measures satisfying the OSC. J. Math. Anal. Appl. 379, 834–841 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Zhou, X., Chen, E.: Packing dimensions of the divergence points of self-similar measures with open set condition. Monatshefte für Mathematik 172, 233–246 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Zhou, X., Chen, E.: The dimension of the divergence points of self-similar measures witn weak separation condition. Monatshefte für Mathematik 183, 379–391 (2017)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Basic Teaching DepartmentNanjing University Jinling CollegeNanjingPeople’s Republic of China
  2. 2.School of Mathematical Sciences and Institute of MathematicsNanjing Normal UniversityNanjingPeople’s Republic of China

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