Abstract
We prove results on absolute continuity and singularity of the distribution of geometric series with randomly increasing exponents.
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References
Björklund M., Schnellmann D.: Almost sure absolute continuity of Bernoulli convolutions. Ann. Inst. H. Poincaré Probab. Statist. 46, 888–893 (2010)
P. Billingsley, Ergodic Theory and Information, John Wiley and Sons, 1965.
N. Bourbaki, Elements of Mathematics: General Topology, Addison-Wesley, 1966.
Dajani K, Fieldsteel A: Equipartition of interval partitions and an application to number theory. Proc. Amer. Math. Soc. 129, 3453–3460 (2001)
K. Falconer, Fractal Geometry - mathematical foundations and applications, John Wiley and Sons, 1990.
A. Fan and J. Zhang, Absolute continuity of the distribution of some Markov geometric series, Science in China, Series A Mathematics, vol. 50 (2007), 1521–1528.
H. Fernau, Infinite Iterated function systems, Mathematische Nachrichten, vol. 170 (1994), 79–91.
Hutchinson J.E.: Fractals and self-similarity. Indiana Univ. Math. J. 30, 271–280 (1981)
Jordan T, Pollicott M, Simon K: Hausdorff dimension of randomly perturbed self-affine attractors. Comm. in Math. Phys. 270, 519–544 (2007)
P. Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge University Press, 1995.
J. Neunhäuserer, Properties of some overlapping self-similar and some self-affine measures, Acta Mathematica Hungarica, vol. 92 (2001), 143–161.
Neunhäuserer J.: A general result on absolute continuity of non-uniform self-similar measures on the real line. Fractals 16, 299–304 (2008)
Y. Peres, W. Schlag, and B. Solomyak, Sixty years of Bernoulli convolutions, Progress in Probability Vol. 46, Birkhauser, 39–65.
Peres Y, Solomyak B: Absolutely continuous Bernoulli convolutions - a simple proof. Math. Research Letters 3, 231–239 (1996)
Peres Y, Solomyak B: Self-similar measures and intersection of Cantor sets, Trans. Amer. Math. Soc 350, 4065–4087 (1998)
Ya. Pesin, Dimension Theory in Dynamical Systems - Contemporary Views and Applications, University of Chicago Press, 1997.
K. Simon and H. Toth, The absolute continuity of the distribution of random sums with digits {0,1, . . . , m−1}, Real Analysis Exchange, vol 30 (2005), 397–410.
Shmerkin P, Solomyak B: Zeros of {−1, 0, 1} power series and connectedness loci of self-affine sets. Experiment. Math. 15, 499–511 (2006)
Toth H.: Infinite Bernoulli convolutions with different probabilities. Dis. Contin. Dyn. Sys. 21, 595–600 (2008)
Young L.S.: Dimension, entropy and Lyapunov exponents. Ergod. Thy. & Dynam. Sys. 2, 109–124 (1982)
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Neunhäuserer, J. Geometric series with randomly increasing exponents. Arch. Math. 102, 283–291 (2014). https://doi.org/10.1007/s00013-014-0621-9
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DOI: https://doi.org/10.1007/s00013-014-0621-9