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Singularity of self-similar measures with respect to Hausdorff measures
Author(s):
Manuel
Morán;
José-Manuel
Rey
Journal:
Trans. Amer. Math. Soc.
350
(1998),
2297-2310.
MSC (1991):
Primary 28A78, 28A80
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Abstract:
Besicovitch (1934) and Eggleston (1949) analyzed subsets of points of the unit interval with given frequencies in the figures of their base- 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.
References:
- [1]
- C. Bandt, Deterministic fractals and fractal measures, in Lecture notes of the School on Measure Theory and Real Analysis, Grado, Italy, 1991, Rend. Ist. Mat. Univ. Trieste 23 (1991), 1-40 (1993). MR 95e:28002
- [2]
- M. Barnsley, Fractals everywhere, Academic Press, 1988. MR 90e:58080
- [3]
- A.S. Besicovitch, On the sum of digits of real numbers represented in the dyadic system, Math. Ann. 110 (1934), 321-330.
- [4]
- P. Billingsley, Hausdorff dimension in probability theory, Illinois J. Math. 4 (1960), 187-209. MR 24:A1750
- [5]
- R. Cawley and R.D. Mauldin, Multifractal decomposition of Moran fractals, Adv. Math. 92 (1992), 196-236. MR 93b:58085
- [6]
- M.J.P. Cooper, The Hausdorff measure of the Besicovitch-Eggleston set, preprint.
- [7]
- A. Deliu, J.S. Geronimo, R. Shonkwiler and D. Hardin, Dimensions associated with recurrent self-similar sets, Math. Proc. Cambridge Phil. Soc. 110 (1991), 327-336. MR 92g:58069
- [8]
- H.G. Eggleston, The fractional dimension of a set defined by decimal properties, Quart. J. Math. Oxford Ser. 20 (1949), 31-36. MR 11:88e
- [9]
- K.J. Falconer, The Multifractal spectrum of statistically self-similar measures, J. Theoret. Probab. 7 (1994), 681-702. MR 95m:60076
- [10]
- J.E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), 713-747. MR 82h:49026
- [11]
- M. Morán, Hausdorff measure of infinitely generated self-similar sets, Monatsh. Math. 122 (1996), 387-399. MR 97j:28015
- [12]
- M. Morán and J.-M. Rey, Geometry of self-similar measures, Ann. Acad. Sci. Fenn. Mathematica 22 (1997), 365-386.
- [13]
- P.A.P. Moran, Additive functions of intervals and Hausdorff measure, Proc. Cambridge Phil. Soc. 42 (1946), 15-23. MR 7:278f
- [14]
- N. Patzschke, Self-conformal multifractal measures, preprint.
- [15]
- F. Przytycki, M. Urba\'{n}ski and A. Zdunik, Harmonic, Gibbs and Hausdorff measures on repellers for holomorphic maps, I, Ann. of Math. 130 (1989), 1-40. MR 91i:58115
- [16]
- M.S. Raghunathan, A proof of Oseledec's multiplicative ergodic theorem, Israel J. Math. 32 (1978) 356-362. MR 81f:60016
- [17]
- C.A. Rogers and S.J. Taylor, Functions continuous and singular with respect to a Hausdorff measure, Mathematika 8 (1961), 1-31. MR 24:A200
- [18]
- A. Schief, Separation properties for self-similar sets, Proc. Amer. Math. Soc. 122 (1994), 111-115. MR 94k:28012
- [19]
- M. Smorodinsky, Singular measures and Hausdorff measures, Israel J. Math. 7 (1969), 203-206. MR 40:3589
- [20]
- C. Tricot, Two definitions of fractional dimension, Math. Proc. Cambridge Phil. Soc. 91 (1982), 57-74. MR 84d:28013
- [21]
- P. Walters, An introduction to ergodic theory, Springer-Verlag, 1982. MR 84e:28017
- [22]
- L.S. Young, Dimension, entropy and Liapunov exponents, Ergodic Theory Dynam. Systems 2 (1982), 109-124. MR 84h:58087
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Additional Information:
Manuel
Morán
Affiliation:
Departamento de Análisis Económico, Universidad Complutense, Campus de Somosaguas, 28223 Madrid. Spain
Email:
ececo06@sis.ucm.es
José-Manuel
Rey
Affiliation:
Departamento de Análisis Económico, Universidad Complutense, Campus de Somosaguas, 28223 Madrid. Spain
Email:
ececo07@sis.ucm.es
DOI:
10.1090/S0002-9947-98-02218-1
PII:
S 0002-9947(98)02218-1
Keywords:
Self--similarity,
Hausdorff measures,
dimension function,
Law of the Iterated Logarithm.
Received by editor(s):
January 17, 1996
Additional Notes:
Research partially supported by {\em Ente Público Puertos del Estado}
Copyright of article:
Copyright
1998,
American Mathematical Society
Forward Citation(s): Information for authors on submitting citations The following works have cited this article Manuel Morán, Problems on self-similar geometry, Fractal Geometry and Stochastics II, Progress in Probability, vol. 46, Birkhäuser Verlag Basel, Switzerland, 2000, pp. 70-93.
Manuel Morán and José-Manuel Rey , Geometry of self-similar measures, Ann. Acad. Sci. Fenn. Mathematica 22 (1997), 365-386. MR 99f:28011
Manuel Morán , Dynamical boundary of a self-similar set , Fund. Math. 160 (1999), 1-14.
José-Manuel Rey , The role of Billingsley dimensions in computing fractal dimensions on Cantor-like spaces, Proc. Amer. Math. Soc. (2) 128 (2000), posted on 07/06/1999, 561-572. MR 2000c:28017
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