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Fractals in the Large

Published online by Cambridge University Press:  20 November 2018

Robert S. Strichartz*
Affiliation:
Mathematics Department White Hall Cornell University Ithaca, NY USA 14853 e-mail: str@math.cornell.edu
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Abstract

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A reverse iterated function system (r.i.f.s.) is defined to be a set of expansive maps $\left\{ {{T}_{1}},...,{{T}_{m}} \right\}$ on a discrete metric space $M$. An invariant set $F$ is defined to be a set satisfying $F\,=\,\bigcup _{j=1}^{m}\,{{T}_{j}}F$, and an invariant measure $\mu $ is defined to be a solution of $\mu \,=\,\sum{_{j=1}^{m}\,{{p}_{j}}\mu {}^\circ T_{j}^{-1}}$ for positive weights ${{p}_{j}}$. The structure and basic properties of such invariant sets and measures is described, and some examples are given. A blowup$\mathcal{F}$ of a self-similar set $F$ in ${{\mathbb{R}}^{n}}$ is defined to be the union of an increasing sequence of sets, each similar to $F$. We give a general construction of blowups, and show that under certain hypotheses a blowup is the sum set of $F$ with an invariant set for a r.i.f.s. Some examples of blowups of familiar fractals are described. If $\mu $ is an invariant measure on ${{\mathbb{Z}}^{+}}$ for a linear r.i.f.s., we describe the behavior of its analytic transform, the power series $\sum{_{n=0}^{\infty }\mu (n){{z}^{n}}}$ on the unit disc.

Keywords

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1998

References

[Ba] Bandt, C., Self-similar tilings and patterns described by mappings, NATO Advanced Study Institute Proceedings on “Long Range Aperiodic Order” (Ed. Moody, R.V.). Kluwer Academic Publisher, 1997, 45-83.Google Scholar
[B] Barnsley, M.F., Fractals Everywhere, Academic Press, Boston, 1988.Google Scholar
[BF] Bedford, T. and Fisher, A., Analogues of the Lebesgue density theorem for fractal sets of reals and integers, Proc. London Math. Soc. (3) 64(1992), 95124.Google Scholar
[Hoi] Holschneider, M., Large scale renormalization of Fourier transforms of self similar measures and selfsimilarity ofRiesz measures, J. Math. Anal. Appl., to appear.Google Scholar
[Hu] Hutchinson, J.E., Fractals and self similarity. Indiana Univ. Math. J. 30(1981), 713747.Google Scholar
[JRS] Janardhan, R., Rosenblum, D. and Strichartz, R.S., Numerical experiments in Fourier asymptotics of Cantor measures and wavelets. Experimental Math. 1(1992), 249-273.Google Scholar
[LI] Lau, K.-S., Fractal measures and mean p-variations. J. Funct. Anal. 108(1992), 427457.Google Scholar
[L2] Lau, K.-S., Dimension of a family of singular Bernoulli convolutions, J. Funct. Anal. 116(1993), 335—358.Google Scholar
[LN] Lau, K.-S. and Ngai, S.-M., Lq-spectrum of the Bernoulli convolutions associated with the Golden ratio, preprint.Google Scholar
[LW] Lau, K.-S. and Wang, J., Mean quadratic variations and Fourier asymptotics of self-similar measures,. Monat. Math. 115(1993), 99-132.Google Scholar
[M] Mandelbrot, B., The fractal geometry of nature, W. H. Freeman and Co., San Francisco, 1982.Google Scholar
[Se] Senechal, M., Quasicrystals and geometry, Cambridge Univ. Press, 1995.Google Scholar
[SI] Strichartz, R.S., Self similar measures and their Fourier transforms I. Indiana Univ. Math. J. 39(1990), 797-817.Google Scholar
[S2] Strichartz, R.S., Self-similar measures and their Fourier transforms II. Trans. Amer. Math. Soc. 336(1993), 335— 361.Google Scholar
[S3] Strichartz, R.S., Self-similar measures and their Fourier transforms III, Indiana Univ. Math. J. 42(1993), 347411.Google Scholar
[S4] Strichartz, R.S., Self similarity in harmonic analysis. J. Fourier Anal. Appl. 1(1994), 1—37.Google Scholar
[STZ] Strichartz, R.S., Taylor, A. and Zhang, T., Densities of self-similar measures on the line. Experimental Math. 4(1995), 101-128.Google Scholar