Fractal surfaces
A compact metric space X with the family F of functions \(f_ i: X\to X,\quad 1\leq i\leq N,\) is said to be an iterated function system (X,F). The family H(X) of closed subsets of X is assumed to be equipped with the Hausdorff metric. Provided the maps \(f_ i\), \(1\leq i\leq N\), are contractive, the set-valued map \(\hat f:\;H(X)\to H(X),\) defined by \(\hat f(A)=\cup^{N}_{i=1}f_ i(A),\) is a contraction on H(X), thus possessing a unique fixed point, called the attractor of (X,F). A fractal surface in \({\mathbb{R}}^ 3\) is defined as an attractor of a certain iterated function system. The components of F may be chosen as \(f_ i(x,y,z)=(\Phi_ i(x,y),\Psi_ i(x,y,z)),\) where \(\Phi_ i:\;[0,1]^ 2\to [0,1]^ 2,\Psi_ i:\;X\to {\mathbb{R}},\quad 1\leq i\leq N,\) \(N\geq 2\), \(X=[0,1]^ 2\times {\mathbb{R}}.\) The functions \(\Phi_ i\) and \(\Psi_ i\) ought to satisfy several conditions, e.g. \(\Psi_ i\) should be Lipschitz in the first and second variable and contractive in the third. In practice the functions \(f_ i\) may be safely thought as affine ones. It was proven that the attractor of (X,F) is the graph of a certain continuous function \(z=\chi (x,y).\) This graph is in general a self- similar fractal set which is referred to a fractal surface. The code space and the dynamical system associated with the fractal surface are also considered. The paper contains materials on the Lyapunov dimension, p-balanced measures and the moment theory, Hölder continuity of fractal surfaces. The fractal dimension is evaluated for a special case of affine generating maps.
scientific article; zbMATH DE number 4179704
scientific article