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Zbl 0598.28011
Hutchinson, John E.
Fractals and self similarity. (English)
[J] Indiana Univ. Math. J. 30, 713-747 (1981). ISSN 0022-2518

The term "fractal" introduced by B. Mandelbrot refers to classes of sets having either strict or statistical self-similarities. The usual Cantor set is a strict example; such sets frequently are Cantor-type sets with nonintegral Hausdorff dimensions. The Cantor set has dimension log 2/log 3. Mandelbrot and others have used such sets extensively to model various physical and biological phenomena. Mandelbrot typically has obtained his examples of strictly self-similar fractals by ad hoc constructions based on "initial" and "standard" polygons and appropriate iterative procedures. It is the present author's thesis (successfully presented, in the opinion of the reviewer) that a better way to regard a fractal is as a finite collection ${\cal S}=\{S\sb 1,...,S\sb N\}$ of contraction mappings; the fractal $\vert {\cal S}\vert$ is then determined by the requirement that $\vert {\cal S}\vert =\cup\sb{i}S\sb i\vert {\cal S}\vert$ ($\vert {\cal S}\vert$ does not necessarily determine ${\cal S}$ uniquely). In Mandelbrot's published examples each ${\cal S}$ would consist of similitudes of ${\bbfR}\sp n$ (the composition of an isometry with a homothety); one can thus readily classify all possible such strictly self-similar fractals and perhaps think in terms of constructing an atlas (P. E. Oppenheimer, in his 1979 Princeton senior thesis obtained a computer generation of an atlas of part of one component of the parameter space, with dramatic results). For the usual Cantor set as above one can take $n=1,$ $N=2,$ and let $S\sb 1$, $S\sb 2$ be orientation-preserving similitudes with 0, 1 as respective fixed points and conraction ratios of 1/3. Among the basic results of the present paper are the following. \par (1) Let X be a complete metric space and ${\cal S}=\{S\sb 1,...,S\sb N\}$ be a finite set of contraction mappings on X. Then there exists a unique closed bounded set $\vert {\cal S}\vert$ such that $\vert {\cal S}\vert =\cup\sb{i}S\sb i\vert {\cal S}\sb i\vert.$ Furthermore, $\vert {\cal S}\vert$ is compact and is the closure of the set of fixed points of finite compositions $S\sb{i(1)}\circ...\circ S\sb{i(p)}$ of members of ${\cal S}$. Furthermore, for arbitrary nonempty closed bounded $A\subset X$, ${\cal S}\sp p(A)\to \vert {\cal S}\vert$ in the Hausdorff metric; here ${\cal S}(A)=\cup\sb{i}S\sb i(A),$ ${\cal S}\sp p(A)={\cal S}({\cal S}\sp{-1}(A)).$ (2) Suppose additionally that $\rho\sb 1,..,\rho\sb N\in (0,1)$ with $\Sigma\sb i\rho\sb i=1.$ Then there is a unique Borel regular measure $\Vert {\cal S},\rho \Vert$ of total mass 1 such that $\Vert {\cal S},\rho \Vert =\Sigma\sb i\rho\sb iS\sb{i\#}\Vert {\cal S},\rho \Vert.$ Furthermore $spt\Vert {\cal S},\rho \Vert =\vert {\cal S}\vert.$ \par Additional results of this paper show in some cases how to associate an m-dimensional integral flat chain to ${\cal S}$ (m an integer) even though $\vert {\cal S}\vert$ is not of integral dimension. The author also examines relationships between similarity dimension and Hausdorff dimension and between $\Vert {\cal S},\rho \Vert$ and Hausdorff measure. Finally he gives conditions guaranteeing that $\vert {\cal S}\vert$ is purely unrectifiable (even though of infinite measure).

MSC 2000:: *28A75 Geometric measure theory
49Q15 Geometric measure and integration theory, etc.
58C25 Differentiable maps on manifolds (global analysis)
58K99 None of the above, but in this section

Keywords: Cantor-type sets with nonintegral Hausdorff dimensions; self-similar fractals; Hausdorff metric; Hausdorff measure

Cited in: Zbl 1246.42002 Zbl 1232.28005 Zbl 1224.26022 Zbl 1190.37007 Zbl 1187.37027 Zbl 1171.28002 Zbl 1148.28006 Zbl 1143.28006 Zbl 1125.28010 Zbl 1115.28008 Zbl 1048.28005 Zbl 1040.28013 Zbl 1034.28005 Zbl 1016.28008 Zbl 1009.60043 Zbl 1073.37506 Zbl 1027.54046 Zbl 0995.28005 Zbl 0980.28005 Zbl 0949.28004 Zbl 0978.28006 Zbl 0959.28004 Zbl 0948.28004 Zbl 0929.28007 Zbl 0929.28005 Zbl 0929.28008 Zbl 0956.28007 Zbl 0881.28001 Zbl 0932.01053 Zbl 0874.54024 Zbl 0866.60065 Zbl 0823.34071 Zbl 0994.54502 Zbl 0817.28006 Zbl 0847.54037 Zbl 0748.68020 Zbl 0741.58029 Zbl 0953.94504 Zbl 0743.60076 Zbl 0739.60038 Zbl 0721.28006 Zbl 0687.30029 Zbl 0779.54022 Zbl 0652.60068 Zbl 0642.28005 Zbl 0652.60084 Zbl 0623.60020 Zbl 0613.28008 Zbl 0613.28007 Zbl 0618.54030

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