Approximating the Attractor Set of Iterated Function Systems of Order m by $$\alpha $$-Dense Curves

Abstract

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 order m. Our main tool are the so-called α-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.

Approximating intuitionistic fuzzy fractals by densifiability techniques

“…, f n }, n > 1, is an IFS in the metric space (X, d), there are some procedures to approximate the attractor set of F; see, for instance, [9,11]. Moreover, the author has proved in [15] (see also [16]) that the α-dense curves can be used to approximate under suitable conditions, with arbitrarily small and controlled error, the attractor set of the so called countable iterated functions systems (see, for instance, [33]) and, consequently, the attractor set of certain IFS. We expose this result in the following lines.…”