Densities of self-similar measures on the line. (English) Zbl 0860.28005

Strichartz, Robert S.; Taylor, Arthur; Zhang, Tong

Exp. Math. 4, No. 2, 101-128 (1995).

This paper mainly deals with a numerical approximation to compute self-similar measures \(\mu\) on \([0,1]\) such that \(\mu= \sum^m_{j=1}p_j\mu\circ S^{-1}_j\) for an IFS \(([0,1]\); \(S_1,\dots,S_m\)) and positive weights \(p_j\), but also with self-replicating measures \(\mu\) such that for Borel sets \(A\) in \([0,1]\), \(\mu(A)= \sum^m_{j=1} \int_{S^{-1}_jA} p_jd\mu\), and the \(p_j\) become nonnegative functions.
A set of interval partition data is a pair \(({\mathcal J},\nu)\) such that \(\nu:{\mathcal J}\to[0,1]\) and \(\mathcal J\) is a finite collection of non-overlapping subintervals \(J\subset [0,1]\) such that \(\sum_{J\in{\mathcal J}}\nu(J)=1\). \(\mu\) matches the data \(({\mathcal J},\nu)\) exactly iff \(\mu(J)=\nu(J)\) for all \(J\in{\mathcal J}\).
The numerical algorithm is as follows: Set \({\mathcal J}_0=\{[0,1]\}\), \(\nu_0([0,1])=1\), and given \(({\mathcal J}_{k-1},\nu_{k-1})\), let \[ {\mathcal J}_k= \{S_jJ\mid j=1,\dots,m,\;J\in {\mathcal J}_{k-1}\}\quad\text{and} \quad \nu_k(S_jJ)= p_j\nu_{k-1}(J)\text{ for } J\in {\mathcal J}_{k-1}. \] Now, if \(\mu\) matches \(({\mathcal J}_{k-1},\nu_{k-1})\), so it will \(({\mathcal J}_k,\nu_k)\). Later on, this simple algorithm will be modified. The authors discuss error measurement of these algorithms. The numerical approach to \(\mu\) is used to study the density diagram of \(\mu\), \[ \{(s,h(x,s))\mid x\in\text{carrier}(\mu),\;s\geq 0\}, \] where \[ h(x,s)= {\mu(B_{c^{-s}}(x))\over (2c^{-s})^\alpha} \] for given \(\alpha,c>0\) and \(B_{c^{-s}}(x)\) the closed ball at \(x\) and radius \(c^{-s}\). The diagram is (highly) self-similar and overcomes the problem of non-existence of an exact density function.
Finally, the paper includes also the problem of the correct normalization of Hausdorff measures and some other averaging density definitions.

Reviewer: H.Haase (Greifswald)

MSC:

28A80

Fractals

Keywords:

iterated function system; self-similar measures; IFS; self-replicating measures; numerical algorithm; density diagram; Hausdorff measures

References:

[1]	Aaronson J., Trans. Amer. Math. Soc. 337 pp 495– (1993) · Zbl 0789.28010 · doi:10.2307/2154231
[2]	Ayer E., ”Exact Hausdorff measure and intervals of maximum density for Cantor sets” (1995)
[3]	Bandt C., Monatsch. Math. 108 pp 89– (1989) · Zbl 0712.58039 · doi:10.1007/BF01308664
[4]	Bandt C., ”The tangential distribution for self-similar measures” (1992)
[5]	Barnsley M., Fractals Everywhere (1988)
[6]	Barnsley M. F., Adv. Appl. Prob. 20 pp 14– (1988) · Zbl 0643.60050 · doi:10.2307/1427268
[7]	Barnsley M. F., Constr. Approx. (1989)
[8]	Bedford T., of reals and integers Proc. London Math. Soc. (3) 64 pp 95– (1992) · Zbl 0706.28009 · doi:10.1112/plms/s3-64.1.95
[9]	Blanchard P., Bull. Amer. Math. Soc. 11 pp 85– (1984) · Zbl 0558.58017 · doi:10.1090/S0273-0979-1984-15240-6
[10]	Bowen R., Equilibrium states and the ergodic theory of Anosov diffeomorphisms (1975) · Zbl 0308.28010 · doi:10.1007/BFb0081279
[11]	Cawley R., Advances in Math. 92 pp 196– (1992) · Zbl 0763.58018 · doi:10.1016/0001-8708(92)90064-R
[12]	Dudley R. M., Studia Math. 27 pp 251– (1966)
[13]	Edgar G. A., Proc. London Math. 65 pp 604– (1992) · Zbl 0764.28007 · doi:10.1112/plms/s3-65.3.604
[14]	Falconer K. J., The geometry of fractal sets (1985) · Zbl 0587.28004 · doi:10.1017/CBO9780511623738
[15]	Falconer K. J., Fractal Geometry: Mathematical Foundations and Applications (1990)
[16]	Falconer K. J., J. Stat. Phys. 67 pp 781– (1992) · Zbl 0893.28006 · doi:10.1007/BF01049726
[17]	Graf S., ”On Bandt’s tangential distribution of self-similar measures” (1993) · Zbl 0841.28011
[18]	Hofbauer F., monotonic maps of the interval Studia Math. 103 pp 191– (1992)
[19]	Hutchinson J. E., Indiana U. Math. J. 30 pp 713– (1981) · Zbl 0598.28011 · doi:10.1512/iumj.1981.30.30055
[20]	Janardhan P., measures and wavelets Experimental Math. pp 249– (1992) · Zbl 0788.65129 · doi:10.1080/10586458.1992.10504561
[21]	Lau K.-S., J. Functional Anal. 108 pp 427– (1992) · Zbl 0767.28007 · doi:10.1016/0022-1236(92)90031-D
[22]	Lau K.-S., J. Functional Anal. 116 pp 335– (1993) · Zbl 0788.60055 · doi:10.1006/jfan.1993.1116
[23]	Lau K.-S., Monat. Math. 115 pp 99– (1993) · Zbl 0778.28005 · doi:10.1007/BF01311213
[24]	Marion J., Ann. sc. math. Québec 10 pp 51– (1986)
[25]	Marion J., Ann. sc. math. Québec 11 pp 111– (1987)
[26]	Patzschke N., Proc. Amer. Math. Soc. 117 pp 137– (1993)
[27]	Rachev S. T., Probability metrics and the stability of stochastic models (1991) · Zbl 0744.60004
[28]	Ruelle D., Scaling and self-similarity in physics: renormalization in statistical mechanics and dynamics pp 351– (1983) · doi:10.1007/978-1-4899-6762-6_11
[29]	Stein E. M., Introduction to Fourier analysis on Euclidean spaces (1971)
[30]	Strichartz R. S., J. Functional Anal. 89 pp 154– (1990) · Zbl 0693.28005 · doi:10.1016/0022-1236(90)90009-A
[31]	Strichartz R. S., Indiana U. Math. J. 39 pp 797– (1990) · Zbl 0695.28003 · doi:10.1512/iumj.1990.39.39038
[32]	Strichartz R. S., J. Functional Anal. 102 pp 176– (1991) · Zbl 0741.28006 · doi:10.1016/0022-1236(91)90140-Z
[33]	Strichartz R. S., Trans. Amer. Math. Soc. 336 pp 335– (1993) · Zbl 0765.28007 · doi:10.2307/2154350
[34]	Strichartz R. S., Indiana U. Math. J. 42 pp 367– (1993) · Zbl 0790.28003 · doi:10.1512/iumj.1993.42.42018
[35]	Strichartz R. S., J. Fourier Anal. Appl. (1994)
[36]	Sullivan D., Geometric Dynamics pp 725– (1983) · doi:10.1007/BFb0061443
[37]	Tsujii Y., Hiroshima Math. J. 21 pp 491– (1991)
[38]	Young L.-S., Ergod. Th. Dynam. Sys. pp 109– (1982) · Zbl 0523.58024 · doi:10.1017/S0143385700009615