Skip to main content
×
Home
    • Aa
    • Aa

Multifractal structure of Bernoulli convolutions

  • THOMAS JORDAN (a1), PABLO SHMERKIN (a2) and BORIS SOLOMYAK (a3)
Abstract
Abstract

Let νpλ be the distribution of the random series , where in is a sequence of i.i.d. random variables taking the values 0, 1 with probabilities p, 1 − p. These measures are the well-known (biased) Bernoulli convolutions.

In this paper we study the multifractal spectrum of νpλ for typical λ. Namely, we investigate the size of the sets Our main results highlight the fact that for almost all, and in some cases all, λ in an appropriate range, Δλ, p(α) is nonempty and, moreover, has positive Hausdorff dimension, for many values of α. This happens even in parameter regions for which νpλ is typically absolutely continuous.

Copyright
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[2] M. Arbeiter and N. Patzschke Random self-similar multifractals. Math. Nachr. 181 (1996), 542.

[3] R. Cawley and R. D. Mauldin Multifractal decompositions of Moran fractals. Adv. Math. 92 (1992), 196236.

[9] D.-J. Feng and H. Hu Dimension theory of iterated function systems. Comm. Pure Appl. Math. 62 (11) (2009), 14351500.

[10] D.-J. Feng and K.-S. Lau Multifractal formalism for self-similar measures with weak separation condition. J. Math. Pures Appl. (9), 92 (4) (2009), 407428.

[11] D.-J. Feng , K.-S. Lau and X.-Y. Wang Some exceptional phenomena in multifractal formalism. II. Asian J. Math. 9 (4), (2005), 473488.

[13] A. M. Garsia Arithmetic properties of Bernoulli convolutions. Trans. Amer. Math. Soc. 102 (1962), 409432.

[14] P. Glendinning and N. Sidorov Unique representations of real numbers in non-integer bases. Math. Res. Lett. 8 (4) (2001), 535543.

[16] V. Komornik and P. Loreti Unique developments in non-integer bases. Amer. Math. Monthly 105 (7) (1998), 636639.

[17] F. Ledrappier On the dimension of some graphs. In Symbolic dynamics and its applications (New Haven, CT, 1991) Contemp. Math, vol. 135, pages 285293 (Amer. Math. Soc., Providence, RI, 1992).

[18] R. D. Mauldin and K. Simon The equivalence of some Bernoulli convolutions to Lebesgue measure. Proc. Amer. Math. Soc. 126 (9) (1998), 27332736.

[19] E. Olivier , N. Sidorov and A. Thomas On the Gibbs properties of Bernoulli convolutions related to β-numeration in multinacci bases. Monatsh. Math., 145 (2): 145174, 2005.

[20] L. Olsen Multifractal Geometry. In Fractal Geometry and Stochastics, II (Greifswald/Koserow, 1998), vol. 46 of Progr. Probab. vol. 46 (Birkhäuser, Basel, 2000), pp. 337.

[21] L. Olsen A lower bound for the symbolic multifractal spectrum of a self-similar multifractal with arbitrary overlaps. Math. Nachr. 282 (10) (2009), 14611477.

[22] W. Parry On the β-expansions of real numbers. Acta Math. Acad. Sci. Hungar. 11 (1960), 401416.

[24] 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. 3965.

[25] Y. Peres and B. Solomyak Absolute continuity of Bernoulli convolutions, a simple proof. Math. Res. Lett. 3 (2) (1996), 231239.

[26] Y. Peres and B. Solomyak Self-similar measures and intersections of Cantor sets. Trans. Amer. Math. Soc. 350 (10) (1998), 40654087.

[27] P. Shmerkin and B. Solomyak Zeros of {−1, 0, 1} power series and connectedness loci for self-affine sets. Experiment. Math. 15 (4) (2006), 499511.

[28] B. Solomyak On the random series ∑ ± λn (an Erdős problem). Ann. of Math. (2), 142 (3) (1995), 611625.

[30] H. R. Tóth Infinite Bernoulli convolutions with different probabilities. Discrete Contin. Dyn. Syst. 21 (2) (2008), 595600.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 5 *
Loading metrics...

Abstract views

Total abstract views: 46 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 30th May 2017. This data will be updated every 24 hours.