PARAMETERIZED CURVE AS ATTRACTORS OF SOME COUNTABLE ITERATED FUNCTION SYSTEMS

Abstract. In this paper we will demonstrate that, in some conditions, the attractor of a countable iterated function system is a parameterized curve. This fact results by generalizing a construction of J. E. Hutchinson [3]. 1. Preliminary facts We will present some notions and results used in the sequel (more complete and rigorous treatments may be found in [2], [1]). 1.1. Hausdor metric. Let (X; d) be a complete metric space and K(X) be the class of all compact non-empty subsets of X. If we dene a function : K(X)K(X) ! R+, (A; B) = maxfd(A; B); d(B; A)g; where d(A; B) = sup x2A ( inf y2B d(x; y)) ; for all A; B 2 K(X); we obtain a metric, namely the Hausdor metric. The set K(X) is a complete metric space with respect to this metric . 1.2. Parameterized curve in the case of iterated function systems. In this section, we will present the iterated functions system (abbreviated IFS) and the Hutchinson's construction of a continuous function f dened to [0; 1] such that Im(f) (the image of f) is the attractor of some IFS (for details see [3]). Let (X; d) be a complete metric space. A set of contractions (!n) N n=1, N 1, is called an iterated function system (IFS), according to M. Barnsley. Such a system of maps induces a set function S: K(X) ! K(X), S(E)