A Note on Invariant Sets of Iterated Function Systems
نویسندگان
چکیده
We prove that the family of all invariant sets of iterated systems of contractions R → R is a nowhere dense Fσ type subset in the space of the non-empty compact subsets of R equipped with the Hausdorff metric. An iterated function system (IFS for short) is a finite collection (T1, . . . , Tn) 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 j X is called an invariant set for the system if A = T1(A)∪ · · · ∪Tn(A). Given a real number 0 < r < 1, a mapping T is called an r-contraction if d (
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