Acta Math. Hungar., 2007 DOI: 10.1007/s10474-007-7013-6 A NOTE ON INVARIANT SETS OF ITERATED FUNCTION SYSTEMS L. L. STACHÓ * and L. I. SZABÓ Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, H-6720 Szeged, Hungary e-mails: stacho@math.u-szeged.hu, lszabo@math.u-szeged.hu (Received January 12, 2007; revised June 22, 2007; accepted July 2, 2007) Abstract. We prove that the family of all invariant sets of iterated systems of contractions R N → R N is a nowhere dense Fσ type subset in the space of the non-empty compact subsets of R N equipped with the Hausdorff metric. An iterated function system (IFS for short) is a finite collection (T 1 ,...,T n ) of weak contractions of a metric space X . By a weak contraction we mean a mapping T : X → X such that d ( T (x),T (y) ) <d(x, y) for all x, y ∈ X , where d is the metric on X . A subset A  X is called an invariant set for the system if A = T 1 (A) ∪···∪ T n (A). Given a real number 0 <r< 1, a mapping T is called an r-contraction if d ( T (x),T (y) ) <r · d(x, y) for all x, y. The term contraction without adjectives refers to r-contraction for some 0 ≦ r< 1 according to the most widespread terminology in the literature. It is known that if the space X is complete then, for any IFS of contractions, there exists a unique nonempty compact invariant set (see [1], [3], [4]). A general IFS may admit no invariant sets, however, it is not hard to see that if it has a compact invariant set then this must be unique. It is also known that any compact set in the euclidean spaces R n can be arbitrarily closely ap- proximated (in Hausdorff distance) by invariant sets of suitably chosen IFSs; * Supported by the Hungarian research grant No. OTKA T/17 48753. Key words and phrases: iterated function system, fractal, invariant set, weak contraction. 2000 Mathematics Subject Classification: 49F20. 0236–5294/$ 20.00 c  2007 Akadémiai Kiadó, Budapest