zbMATH — the first resource for mathematics

Fractal geometry. (Czech) Zbl 0704.28004
The paper contains a short information about fractals, i.e. about sets whose Hausdorff dimension is higher than their topological dimension. The most simple fractals are self-similar sets, i.e. sets in \(E_ n\), for which there exist maps \(\phi_ 1,...,\phi_ m:\;E_ n\to E_ n,\quad m\geq 2,\) such that \(E=\cup^{m}_{i=1}\phi_ i(E),\) where \(\phi_ i\) are compositions of isometries and homotheties and where the intersections \(\phi_ i(E)\cap \phi_ j(E)\) for \(i\neq j\) are not “too big”. Another example of fractals are Julia’s sets. Let \(f_ c(z)=z^ 2+c,\) where c is a complex number. The Julia’s set \(J_ c\) of the function \(f_ c(z)\) is the boundary of the set of complex numbers z for which the sequence \(f_ c(z),f_ c(f_ c(z)),f_ c(f_ c(f_ c(z))),...\) is bounded.
Many fractals can be generated by support of computer graphics.
Reviewer: M.Jůza
28A80 Fractals
28A78 Hausdorff and packing measures