Approximating the Attractor Set of Iterated Function Systems of Order m by α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}-Dense Curves

We construct a sequence of finite sets which approximates, with arbitrarily small and controlled error, the attractor set of a (finite or countable) given iterated function system of orderm. Our main tool are the so-called α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}-dense curves which are from the Hausdorff–Pompeiu distance point of view, a generalization of the space-filling curves. Also we prove that, under suitable conditions, our result can be used to approximate infinite-dimensional fractals generated by integral equations of Volterra type.