# zbMATH — the first resource for mathematics

Singularity of self-similar measures with respect to Hausdorff measures. (English) Zbl 0899.28002
This paper is motivated by the works of A. S. Besicovitch [Math. Ann. 110, 321-330 (1934; Zbl 0009.39503)] and H. G. Eggleston [Q. J. Math., Oxf. Ser. 20, 31-36 (1949; Zbl 0031.20801)] on the subsets of points of the unit interval with given frequencies in the figures of their base-$$p$$ expansion. The authors extend the analysis to self-similar sets in $$\mathbb R^N$$ and introduce the so-called normal Besicovitch sets by replacing the frequencies of figures with the frequencies of the generating similitudes. Their objective is to analyze the relationships among the normal Besicovitch sets, self-similar measures, and Hausdorff measures. The authors characterize those $$\phi$$-Hausdorff measures with respect to which a self-similar measure is singular or absolutely continuous. They also calculate the Hausdorff dimensions of the normal Besicovitch sets which generalize the results of Besicovitch and Eggleston, and show that the self-similar measures are concentrated on sets whose frequencies of similitudes obey the law of the iterated logarithm.

##### MSC:
 28A78 Hausdorff and packing measures 28A80 Fractals
Full Text:
##### References:
 [1] Christoph Bandt, Deterministic fractals and fractal measures, Rend. Istit. Mat. Univ. Trieste 23 (1991), no. 1, 1 – 40 (1993). School on Measure Theory and Real Analysis (Grado, 1991). · Zbl 0791.28005 [2] Michael Barnsley, Fractals everywhere, Academic Press, Inc., Boston, MA, 1988. · Zbl 0691.58001 [3] A.S. Besicovitch, On the sum of digits of real numbers represented in the dyadic system, Math. Ann. 110 (1934), 321-330. · Zbl 0009.39503 [4] Patrick Billingsley, Hausdorff dimension in probability theory, Illinois J. Math. 4 (1960), 187 – 209. · Zbl 0098.10602 [5] Robert Cawley and R. Daniel Mauldin, Multifractal decompositions of Moran fractals, Adv. Math. 92 (1992), no. 2, 196 – 236. · Zbl 0763.58018 · doi:10.1016/0001-8708(92)90064-R · doi.org [6] M.J.P. Cooper, The Hausdorff measure of the Besicovitch-Eggleston set, preprint. [7] Anca Deliu, J. S. Geronimo, R. Shonkwiler, and D. Hardin, Dimensions associated with recurrent self-similar sets, Math. Proc. Cambridge Philos. Soc. 110 (1991), no. 2, 327 – 336. · Zbl 0742.28002 · doi:10.1017/S0305004100070407 · doi.org [8] H.G. Eggleston, The fractional dimension of a set defined by decimal properties, Quart. J. Math. Oxford Ser. 20 (1949), 31-36. · Zbl 0031.20801 [9] K. J. Falconer, The multifractal spectrum of statistically self-similar measures, J. Theoret. Probab. 7 (1994), no. 3, 681 – 702. · Zbl 0805.60034 · doi:10.1007/BF02213576 · doi.org [10] John E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), no. 5, 713 – 747. · Zbl 0598.28011 · doi:10.1512/iumj.1981.30.30055 · doi.org [11] M. Moran, Hausdorff measure of infinitely generated self-similar sets, Monatsh. Math. 122 (1996), no. 4, 387 – 399. · Zbl 0862.28008 · doi:10.1007/BF01326037 · doi.org [12] M. Morán and J.-M. Rey, Geometry of self-similar measures, Ann. Acad. Sci. Fenn. Mathematica 22 (1997), 365-386. · Zbl 0890.28005 [13] P.A.P. Moran, Additive functions of intervals and Hausdorff measure, Proc. Cambridge Phil. Soc. 42 (1946), 15-23. · Zbl 0063.04088 [14] N. Patzschke, Self-conformal multifractal measures, preprint. · Zbl 0912.28007 [15] Feliks Przytycki, Mariusz Urbański, and Anna Zdunik, Harmonic, Gibbs and Hausdorff measures on repellers for holomorphic maps. I, Ann. of Math. (2) 130 (1989), no. 1, 1 – 40. · Zbl 0703.58036 · doi:10.2307/1971475 · doi.org [16] M. S. Raghunathan, A proof of Oseledec’s multiplicative ergodic theorem, Israel J. Math. 32 (1979), no. 4, 356 – 362. · Zbl 0415.28013 · doi:10.1007/BF02760464 · doi.org [17] C. A. Rogers and S. J. Taylor, Functions continuous and singular with respect to a Hausdorff measure., Mathematika 8 (1961), 1 – 31. · Zbl 0145.28701 · doi:10.1112/S0025579300002084 · doi.org [18] Andreas Schief, Separation properties for self-similar sets, Proc. Amer. Math. Soc. 122 (1994), no. 1, 111 – 115. · Zbl 0807.28005 [19] M. Smorodinsky, Singular measures and Hausdorff measures, Israel J. Math. 7 (1969), 203 – 206. · Zbl 0186.49801 · doi:10.1007/BF02787612 · doi.org [20] Claude Tricot Jr., Two definitions of fractional dimension, Math. Proc. Cambridge Philos. Soc. 91 (1982), no. 1, 57 – 74. · Zbl 0483.28010 · doi:10.1017/S0305004100059119 · doi.org [21] Peter Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York-Berlin, 1982. · Zbl 0475.28009 [22] Lai Sang Young, Dimension, entropy and Lyapunov exponents, Ergodic Theory Dynamical Systems 2 (1982), no. 1, 109 – 124. · Zbl 0523.58024
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.