Publication: Singularity of self-similar measures with respect to Hausdorff measures
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1995
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Facultad de Ciencias Económicas y Empresariales. Decanato
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Abstract
Besicoviteh (1941) and Egglestone (1949) analyzed subsets of points of the unit interval with given frequencies in the figures of their base-p expansions. We extend this analysis to self-similar sets, by replacing the frequencies of figures with the frequencies of the generating similitudes. We focus on the interplay among such sets, self-similar measures, and Hausdorff measures. We give a fine-tuned classification of the Hausdorff measures according to the singularity of the self-similar measures with respect to those measures. We show that the self-similar measures are concentrated on sets whose frequencies of similitudes obey the law of the iterated logarithm
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Bandt, C., Deterministic Fractals and Fractal Measures, Lecture Notes of the School on Measure Theory and Real Analysis, Grado, Italy, 1991.
Barnsley, M., Fractals Everywhere, Academic Press, 1988.
Besicovitch, A.S., "On the Sum of Digits of Real Numbers Represented in the Dyadic System", Math. Annalen, 110 (1934), 321-30.
Billingsley, P., Hausdorff Dimension in Probability Theory, Illinois J. Math. 4 (1960), 187-209.
Deliu, A.; Geronimo, J.S.; Shonkwiler, R. and Hardin, D., Dimensions Associated with Recurrent Self-similar Sets, Math. Proc. Cambo Phil. Soc. 110 (1991), 327-36.
Egglestone, H.G., The Fractional Dimension of a Set Defined by Decimal Properties, Quart. J. of Math. Oxford Ser. 20 (1949), 31-6.
Hutchinson, J.E., Fractals and self similarity, Indiana Univ. Math. J. 30 (1981), 713-47.
Martín, M.A. and Mattila, P., k-Dimensional Regularity Classifications for s-Fractals, Trans. Amer. Math. Soc. 305 (1988), 293-315.
Morán, M., Self-Similar Sets Generated by Infinite Systems of Similitudes, preprint.
Raghunathan, M.S., A Proof of Oseledec's Multiplicative Ergodic Theorem, Israel J. Math., 32, 4 (1978) 356-362.
Rogers, C.A. and S.J. Taylor, Functions Continuous and Singular with respect to a Hausdorff Measure, Mathematika, 8 (1961), 1-31.
Schief, A., SOSC and OSC are Equivalent, preprint.
Tricot, C., Two Definitions of Fractional Dimension, Math. Prac. Cambo Phil. Soc. 91 (1982), 57-74.
Walters, P., An Introduction to Ergodic Theory, Springer-Verlag, 1982.
Young, L.S., Dimension, Entropy and Liapunov Exponents, Ergodic Theory & Dynamical Systems 2 (1982), 109-24.