A note on stable iterated function systems

Let X denote a compact metric space with metric d, and let f : X → X denote a continuous self-map on X . For any subset E of X , we let Cl(E) denote the closure of E. Following [1], we denote by { ,X} an iterated function system, or IFS, onX . That is, is a finite family { f1, . . . , fm} of continuous self-maps on X . In this paper we do not consider the case in which is an infinite family. Let ∞ denote the family of infinite sequences of compositions of functions in . That is, a given sequence F in ∞ is composed of arbitrary compositions of elements of , so that for each positive integer n, Fn = f1 ◦ f2 ◦ ··· ◦ fn, where each fi, 1≤ i≤ n, is an element of { ,X}. An IFS { ,X} is hyperbolic if there is a real number λ, where 0 < λ < 1, such that d( f (x), f (y))≤ λ∗ d(x, y) for all points x and y in X and for all f ∈ . An IFS { ,X} is stable if for every infinite sequence F, diam(Fn(X))→ 0 as n→∞. Equivalently, { ,X} is stable if {Fn(X)}n≥1 has a one-point set as its limit. If { ,X} consists of a single function f , then { ,X} is stable if diam( f k(X))→ 0 as k→∞, where f k is the k-fold composition of f . By the definition of stability, each sequence Fλ in ∞, where λ∈Λ for some indexing set Λ, converges to a unique point xλ. Let F denote the sequence of nested decreasing compact sets formed by the family of sequences {Fλ n(X)}λ∈Λ. For example, in the wellknown IFS { f1, f2, [0,1]}, where f1(x)= (1/3)x and f2(x)= (1/3)x +2/3, F is the nested sequence of closed subintervals of [0,1] converging to the Cantor ternary set. An IFS { ,X} is conjugate to an IFS { ′,X ′} if there exists a homeomorphism h : X → X ′ such that ′ = h◦ ◦h−1 = {h◦ f ◦h−1 : f ∈ }. The conjugacy then assures that the qualitative dynamic behavior of the systems and ′ are largely the same. Clearly, stable systems exist which are not themselves hyperbolic, but which are conjugate to hyperbolic systems. In [1] it is shown that functions which are either continuous and monotone or