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Absolute continuity of complex Bernoulli convolutions

Published online by Cambridge University Press:  16 May 2016

PABLO SHMERKIN  and
BORIS SOLOMYAK
PABLO SHMERKIN
Affiliation:
Department of Mathematics and Statistics, Torcuato Di Tella University, and CONICET, Buenos Aires, Argentina. e-mail: [email protected]
BORIS SOLOMYAK
Affiliation:
Department of Mathematics, Bar Ilan University, Ramat–Gan, 5290002, Israel. e-mail: [email protected]
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Abstract

We prove that complex Bernoulli convolutions are absolutely continuous in the supercritical parameter region, outside of an exceptional set of parameters of zero Hausdorff dimension. Similar results are also obtained in the biased case, and for other parametrised families of self-similar sets and measures in the complex plane, extending earlier results.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 161 , Issue 3 , November 2016 , pp. 435 - 453
Copyright
Copyright © Cambridge Philosophical Society 2016 

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References

REFERENCES

[1] Bandt, C. On the Mandelbrot set for pairs of linear maps. Nonlinearity 15 (4) (2002), 1127–1147.Google Scholar
[2] Bandt, C. and Graf, S. Self-similar sets. VII. A characterization of self-similar fractals with positive Hausdorff measure. Proc. Amer. Math. Soc. 114 (4) (1992), 995–1001.Google Scholar
[3] Barnsley, M. F. and Harrington, A. N. A Mandelbrot set for pairs of linear maps. Phys. D. 15 (3) (1985), 421–432.Google Scholar
[4] Beaucoup, F., Borwein, F., Boyd, D. W. and Pinner, C. Multiple roots of [-1,1] power series. J. Lond. Math. Soc. 57 (1998), 135–147.Google Scholar
[5] Broomhead, D., Montaldi, J. and Sidorov, N. Golden gaskets: variations on the Sierpiński sieve. Nonlinearity 17 (4) (2004), 1455–1480.Google Scholar
[6] Calegary, D., Koch, S. and Walker, A. Roots, Schottky semigroups, and a proof of Bandt's conjecture. Preprint. arXiv:1410.8542.Google Scholar
[7] Gustavo, C. Moreira, T. de A. and Yoccoz, J.-C. Stable intersections of regular Cantor sets with large Hausdorff dimensions. Ann. of Math. (2) 154 (1) (2001), 45–96.Google Scholar
[8] Erdős, P. On a family of symmetric Bernoulli convolutions. Amer. J. Math. 61 (1939), 974–976.Google Scholar
[9] Erdős, P. On the smoothness properties of a family of Bernoulli convolutions. Amer. J. Math. 62 (1940), 180–186.Google Scholar
[10] Grafakos, L. Modern Fourier Analysis. Graduate Texts in Math., vol. 250 (Springer, New York, second edition, 2009).Google Scholar
[11] Hare, K. and Sidorov, N. Two-dimensional self-affine sets with interior points, and the set of uniqueness. Nonlinearity 29 (1) (2016), 1–26.Google Scholar
[12] Hochman, M. On self-similar sets with overlaps and sumset phenomena for entropy in ℝ d . Preprint. arXiv:1503.09043.Google Scholar
[13] Hochman, M. and Shmerkin, P. Local entropy averages and projections of fractal measures. Ann. of Math. (2) 175 (3) (2012), 1001–1059.Google Scholar
[14] Jordan, T. Dimension of fat Sierpiński gaskets. Real Anal. Exchange. 31 (1) (2005/06), 97–110.Google Scholar
[15] Jordan, T. and Pollicott, M. Properties of measures supported on fat Sierpinski carpets. Ergodic Theory Dynam. Systems 26 (3) (2006), 739–754.Google Scholar
[16] Kahane, J.-P. Sur la distribution de certaines séries aléatoires. In Colloque de Théorie des Nombres (Univ. Bordeaux, Bordeaux, 1969), pages 119–122. Bull. Soc. Math. France, Mém. No. 25, Soc. Math. France (Paris, 1971).Google Scholar
[17] Orponen, T. On the distance sets of self-similar sets. Nonlinearity 25 (6) (2012), 1919–1929.Google Scholar
[18] Peres, Y. and Schlag, W. Smoothness of projections, Bernoulli convolutions and the dimension of exceptions. Duke Math. J. 102 (2) (2000), 193–251.Google Scholar
[19] Peres, Y., Schlag, W. and Solomyak, B. Sixty years of Bernoulli convolutions. In Fractal Geometry and Stochastics, II (Greifswald/Koserow, 1998), Progr. Probab. vol. 46 (Birkhäuser, Basel, 2000), pp. 39–65.Google Scholar
[20] Peres, Y. and Shmerkin, P. Resonance between Cantor sets. Ergodic Theory Dynam. Systems 29 (1) (2009), 201–221.Google Scholar
[21] Shmerkin, P. On the exceptional set for absolute continuity of Bernoulli convolutions. Geom. Funct. Anal. 24 (3) (2014), 946–958.Google Scholar
[22] Shmerkin, P. and Solomyak, B. Absolute continuity of self-similar measures, their projections and convolutions. Trans. Amer. Math. Soc. 368 (7) (2016), 5125–5151.Google Scholar
[23] Solomyak, B. On the random series ∑±λ n (an Erdős problem). Ann. of Math. (2) 142 (3) (1995), 611–625.Google Scholar
[24] Solomyak, B. and Xu, H. On the ‘Mandelbrot set’ for a pair of linear maps and complex Bernoulli convolutions. Nonlinearity 16 (5) (2003), 1733–1749.Google Scholar