Fractals and self similarity.

*(English)*Zbl 0598.28011The 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 \({\mathcal S}=\{S_ 1,...,S_ N\}\) of contraction mappings; the fractal \(| {\mathcal S}|\) is then determined by the requirement that \(| {\mathcal S}| =\cup_{i}S_ i| {\mathcal S}|\) (\(| {\mathcal S}|\) does not necessarily determine \({\mathcal S}\) uniquely). In Mandelbrot’s published examples each \({\mathcal S}\) would consist of similitudes of \({\mathbb{R}}^ n\) (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 \(S_ 1\), \(S_ 2\) 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.

(1) Let X be a complete metric space and \({\mathcal S}=\{S_ 1,...,S_ N\}\) be a finite set of contraction mappings on X. Then there exists a unique closed bounded set \(| {\mathcal S}|\) such that \(| {\mathcal S}| =\cup_{i}S_ i| {\mathcal S}_ i|.\) Furthermore, \(| {\mathcal S}|\) is compact and is the closure of the set of fixed points of finite compositions \(S_{i(1)}\circ...\circ S_{i(p)}\) of members of \({\mathcal S}\). Furthermore, for arbitrary nonempty closed bounded \(A\subset X\), \({\mathcal S}^ p(A)\to | {\mathcal S}|\) in the Hausdorff metric; here \({\mathcal S}(A)=\cup_{i}S_ i(A),\) \({\mathcal S}^ p(A)={\mathcal S}({\mathcal S}^{-1}(A)).\) (2) Suppose additionally that \(\rho_ 1,..,\rho_ N\in (0,1)\) with \(\Sigma_ i\rho_ i=1.\) Then there is a unique Borel regular measure \(\| {\mathcal S},\rho \|\) of total mass 1 such that \(\| {\mathcal S},\rho \| =\Sigma_ i\rho_ iS_{i\#}\| {\mathcal S},\rho \|.\) Furthermore \(spt\| {\mathcal S},\rho \| =| {\mathcal S}|.\)

Additional results of this paper show in some cases how to associate an m-dimensional integral flat chain to \({\mathcal S}\) (m an integer) even though \(| {\mathcal S}|\) is not of integral dimension. The author also examines relationships between similarity dimension and Hausdorff dimension and between \(\| {\mathcal S},\rho \|\) and Hausdorff measure. Finally he gives conditions guaranteeing that \(| {\mathcal S}|\) is purely unrectifiable (even though of infinite measure).

(1) Let X be a complete metric space and \({\mathcal S}=\{S_ 1,...,S_ N\}\) be a finite set of contraction mappings on X. Then there exists a unique closed bounded set \(| {\mathcal S}|\) such that \(| {\mathcal S}| =\cup_{i}S_ i| {\mathcal S}_ i|.\) Furthermore, \(| {\mathcal S}|\) is compact and is the closure of the set of fixed points of finite compositions \(S_{i(1)}\circ...\circ S_{i(p)}\) of members of \({\mathcal S}\). Furthermore, for arbitrary nonempty closed bounded \(A\subset X\), \({\mathcal S}^ p(A)\to | {\mathcal S}|\) in the Hausdorff metric; here \({\mathcal S}(A)=\cup_{i}S_ i(A),\) \({\mathcal S}^ p(A)={\mathcal S}({\mathcal S}^{-1}(A)).\) (2) Suppose additionally that \(\rho_ 1,..,\rho_ N\in (0,1)\) with \(\Sigma_ i\rho_ i=1.\) Then there is a unique Borel regular measure \(\| {\mathcal S},\rho \|\) of total mass 1 such that \(\| {\mathcal S},\rho \| =\Sigma_ i\rho_ iS_{i\#}\| {\mathcal S},\rho \|.\) Furthermore \(spt\| {\mathcal S},\rho \| =| {\mathcal S}|.\)

Additional results of this paper show in some cases how to associate an m-dimensional integral flat chain to \({\mathcal S}\) (m an integer) even though \(| {\mathcal S}|\) is not of integral dimension. The author also examines relationships between similarity dimension and Hausdorff dimension and between \(\| {\mathcal S},\rho \|\) and Hausdorff measure. Finally he gives conditions guaranteeing that \(| {\mathcal S}|\) is purely unrectifiable (even though of infinite measure).

##### MSC:

28A75 | Length, area, volume, other geometric measure theory |

49Q15 | Geometric measure and integration theory, integral and normal currents in optimization |

58C25 | Differentiable maps on manifolds |

58K99 | Theory of singularities and catastrophe theory |