Acta Mathematica Hungarica

, Volume 159, Issue 2, pp 563–588 | Cite as

The entropy of Cantor-like measures

  • K. E. Hare
  • K. G. HareEmail author
  • B. P. M. Morris
  • W. Shen


By a Cantor-like measure we mean the unique self-similar probability measure \(\mu\) satisfying \(\mu = \sum^{m-1}_{i=0} p_{i}{\mu} {\circ} S^{-1}_{i}\) where \(S_{i}(x) = \frac{x}{d} + \frac{i}{d} \cdot \frac{d-1}{m-1}\) for integers \(2 \leq d < m \leq 2d - 1\) and probabilities \(p_{i} > 0, {\sum}p_{i} = 1\). In the uniform case \((p_{i} = 1/m {\rm for all} i)\) we show how one can compute the entropy and Hausdorff dimension to arbitrary precision. In the non-uniform case we find bounds on the entropy.

Key words and phrases

entropy Cantor measure Hausdorff dimension 

Mathematics Subject Classification

28A78 28A80 28D20 


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  1. 1.
    S. Akiyama, D.-J. Feng, T. Kempton and T. Persson, On the Hausdorff dimension of Bernoulli convolutions, Int. Math. Res. Notices (to appear),
  2. 2.
    J. C. Alexander and D. Zagier, The entropy of a certain infinitely convolved Bernoulli measure, J. London Math. Soc. (2), 44 (1991), 121–134MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bezhaeva, Z.I., Oseledets, V.I.: The entropy of the Erdős measure for the pseudogolden ratio. Theory Probab. Appl. 57, 135–144 (2013)MathSciNetCrossRefGoogle Scholar
  4. 4.
    E. Breuillard and P. P. Varjú, Entropy of Bernoulli convolutions and uniform exponential growth for linear groups, J. Anal. Math. (to appear), arXiv:1510.04043
  5. 5.
    Bruggeman, C., Hare, K.E., Mak, C.: Multifractal spectrum of self-similar measures with overlap. Nonlinearity 27, 227–256 (2014)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Edson, M.: Calculating the numbers of representations and the Garsia entropy in linear numeration systems. Monatsh. Math. 169, 161–185 (2013)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Erdős, P.: On a family of symmetric Bernoulli convolutions. Amer. J. Math. 61, 974–976 (1939)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Erdős, P.: On the smoothness properties of a family of Bernoulli convolutions. Amer. J. Math. 62, 180–186 (1940)MathSciNetCrossRefGoogle Scholar
  9. 9.
    K. Falconer, Techniques in Fractal Geometry, Wiley and Sons (Chichester, 1997)Google Scholar
  10. 10.
    Feng, D.-J.: The limited Rademacher functions and Bernoulli convolutions associated with Pisot numbers. Adv. Math. 195, 24–101 (2005)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Garsia, A.: Entropy and singularity of infinite convolutions. Pacific J. Math. 13, 1159–1169 (1963)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Grabner, P.J., Kirschenhofer, P., Tichy, R.F.: Combinatorial and arithmetical properties of linear numeration systems. Combinatorica 22, 245–267 (2002)MathSciNetCrossRefGoogle Scholar
  13. 13.
    K.E. Hare, K.G. Hare and M. K-S. Ng, Local dimensions of measures of finite type. II – measures without full support and with non-regular probabilities, Canad. J. Math., 70 (2018), 824–867MathSciNetCrossRefGoogle Scholar
  14. 14.
    Hochman, M.: On self-similar sets with overlaps and inverse theorems for entropy. Ann. of Math. 180, 773–822 (2014)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Jessen, B., Wintner, A.: Distribution functions and the Riemann zeta function. Trans. Amer. Math. Soc. 38, 48–88 (1935)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Lau, K.-S., Ngai, S.-M.: Second-order self-similar identities and multifractal decompositions. Indiana Univ. Math. J. 49, 925–972 (2000)MathSciNetCrossRefGoogle Scholar
  17. 17.
    K.-S. Lau and X.-Y. Wang, Some exceptional phenomena in multifractal formalism. Part I, Asian J. Math., 9 (2005), 275–294MathSciNetCrossRefGoogle Scholar
  18. 18.
    Ngai, S.-M.: A dimension result arising from the \(L^{q}\)-spectrum of a measure. Proc. Amer. Math. Soc. 125, 2943–2951 (1997)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Y. Peres, W. Schlag, and B. Solomyak, Sixty years of Bernoulli convolutions, in: Fractal geometry and stochastics, II (Greifswald/Koserow, 1998), Progr. Probab., Vol. 46, Birkhäuser (Basel, 2000), pp. 39–65Google Scholar
  20. 20.
    Shmerkin, P.: A modified multifractal formalism for a class of self-similar measures with overlap. Asian. J. Math. 9, 323–348 (2005)MathSciNetCrossRefGoogle Scholar
  21. 21.
    P. P. Varjú, Recent progress on Bernoulli convolutions, in: Proceedings of the 7th ECM, Berlin (to appear), arXiv:1608.042Google Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2019

Authors and Affiliations

  • K. E. Hare
    • 1
  • K. G. Hare
    • 1
    Email author
  • B. P. M. Morris
    • 2
  • W. Shen
    • 1
  1. 1.Dept. of Pure MathematicsUniversity of WaterlooWaterlooCanada
  2. 2.Dept. of MathematicsStanford UniversityStanfordUSA

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