On the continuity of the Hutchinson operator

We investigate whether the Hutchinson operator associated with the iterated function system (IFS) is continuous. It clarifies several partial results scattered across recent literature. While the main example for IFS with strict attractor was provided by the family of contractions (the so-called hyperbolic system), the accent was put on ensuring that various contractivity-like conditions are preserved when the Hutchinson operator is induced, unless very recently it was discovered that strict attractors are quite often present for a large class of noncontractive maps, namely projective maps. This sets substantial motivation for the study whether in general continuity of functions guarantees continuity of the induced Hutchinson operator.


Hyperspaces and multifunctions
We shall assume throughout the paper that (X, d) stands for the complete metric space with metric d. The closure of B ⊂ X will be denoted by B. The distance from point b ∈ X to set C ⊂ X is  The family of all nonempty subsets of X is denoted P(X), and becomes an infinite-valued semimetric space when endowed with h (i.e. h(B, C) = ∞ is allowed when at least one of the sets B, C is unbounded, and h(B, C) = 0 implies only that B = C). The hyperspace of compacta is a metric space (K(X), h), where K(X) consists of nonempty compact subsets of X. It is complete, since constructing the hyperspace (of closed sets) preserves completness and precompact if the base space X is so (e.g. [HuPa 1997, IlNa 1999).
Some handy properties of neighborhoods and the Hausdorff distance are collected below.
One striking property of compacta we need later is the following well-known but very usefull fact from general topology.
Any map ϕ : X → P(X) shall be called a multifunction. The image of ∅ = B ⊂ X under ϕ is given by It is the image of a set via relation ϕ ⊂ X × X and it should not be confused with the usual image of map Obviously ϕ(B) = R ϕ (B). Additionally there holds relation ϕ({b}) = ϕ(b) between the image and value of multifunction at b ∈ B, which also should not lead to ambiguity.
If the multifunction ϕ : X → P(X) has compact values, then it is written as ϕ : The set-theoretic union of multifunctions ϕ i : X → P(X), i ∈ I, is defined as i∈I ϕ i : X → P(X), i∈I ϕ i (x) := i∈I ϕ i (x) for x ∈ X. The closure of multifunction ϕ : X → P(X) is defined as ϕ : We can speak about (uniform) continuity and contractivity of the multifunction ϕ : X → P(X) if we equip P(X) with the Hausdorff distance h. For the readers convenience we recall (after [HuPa 1997, Be 1993, AuCe 1984]) these definitions.
Directly from the definition it follows that η fulfills also: η(t) < t, lim sup r→t η(r) < t for t > 0 (e.g. [AnFiGaLe 2005]); see [Ja 1997, Mat 1993 for further discussion of conditions put on comparison functions. Remark 1. The above continuities are meant in the Hausdorff sense and in general should be distincted from continuities in the Vietoris sense. Multivalued contractions are sometimes called Nadler contractions. △ We have the following hierarchy: One also should remember that upper semicontinuity is a notion different from the one used in real analysis; hence upper hemicontinuity has been also coined for multifunctions but in turn to make things worse it is often reserved for multifunctions which are upper semicontinuous w.r.t. the weak topology, so we stack here with a more wide spread term. Upper semicontinuity is essentially a multivalued concept, since for f : X → X and {f } : X → P(X) it holds that {f } is upper semicontinuous if and only if it is continuous which is equivalent to continuity of f .
We remind that the above continuity conditions are preserved under taking the closure and finite union of multifunctions (which can be for example deduced from Proposition 1). It is extended by Theorem 1 to infinite families which are compact.
To be able to deal with compact infinite families of maps as generalization of finite ones we recall after [Le 2004] the notion of the space of multifunctions. The function space M(X, X) of multifunctions ϕ : X → P(X) shall be equipped with an infinite-valued semimetric, namely the Chebyshev distance of uniform convergence for ϕ 1 , ϕ 2 ∈ M(X, X). Of course h sup (ϕ 1 , ϕ 2 ) < ∞ for bounded multifunctions ϕ i : X → P(X), i = 1, 2, i.e. ϕ i (X) -bounded, and h sup (ϕ 1 , ϕ 2 ) = 0 implies that ϕ 1 = ϕ 2 . Hence the subspace of bounded (multi)functions (with closed values) constitutes standard metric function space.
Whenever we speak about (pre)compactness of a family of multifunctions we view it in (M(X, X), h sup ).

Iterated function systems
The system (X, f i : i ∈ I), consisting of a family of maps f i : X → X, will be called iterated function system, shortly IFS on X. When I is finite we speak about finite IFS. Another interesting instance is the compact family of maps (compactness understood in the space of functions with the uniform metric). One can generalize this notion to multivalued IFS (e.g. . This is the main drawback in analyzing the structure of fractals generated by multifunctions. The Hutchinson operator F : P(X) → P(X), associated with a system given by multifunction ϕ : X → P(X) is defined as for B ∈ P(X). In the case of IFS (X, f i : i ∈ I) this means that The n-fold composition of F is written as F n . Whenever we speak about abstract IFS, possibly multivalued, the letter F denotes its associated Hutchinson operator.
The most important instance of the Hutchinson operator is its restriction to the hyperspace of compacta F : K(X) → K(X), since usually the hyperspace K(X) is perceived as habitat for fractals generated by IFSs. To be more precise one has to assume that F sends compacta onto compacta. Indeed this is fulfilled, when the system {f i } i∈I consists of continuous maps and is finite, or more generally compact (in the space of functions equipped with the metric of uniform convergence). Still more general condition can be provided for multivalued IFSs.
Proposition 2. Let ϕ : X → K(X) be an upper semicontinuous multifunction with compact values. Then the induced Hutchinson operator F : P(X) → P(X) transforms compacta into compacta. In particular the restriction F : K(X) → K(X) is well-defined and F (B) = ϕ(B) for B ∈ K(X).
Proof. It is well known that under our assumptions the image of a compact set is again compact ([HuPa 1997, Be 1993]). ⊠ Observe yet that ϕ and its closure ϕ yield the same operator Hutchinson operator ( b∈B ϕ(b) = b∈B ϕ(b) for B ∈ P(X)), so one can always assume that the values of considered multifunctions are closed sets.  Proof. Fix ε > 0. By Proposition 2 in [Le 2003] (comp. Lemma 3 further) we know that for some δ > 0

Attractors and continuity
From the definition of strict attractor there exists n 0 s.t. h(F n (A), A) < δ for n ≥ n 0 , so Combining (4) and (5) gives for n ≥ n 0 and since ε was arbitrary A ⊂ F (A). Due to monotonicity of F then We want to emphasize that if the Hutchinson operator F is continuous, then the invariance of the strict attractor is immediate: This great simplification may be added to a list of typical general interest arguments supporting the search for conditions under which F is continuous. It is known that even very simple upper semicontinuous multifunctions on compact spaces need not induce continuous Hutchinson operator ( Let ϕ : X → P(X) be a multifunction and F : P(X) → P(X) its associated Hutchinson operator. We gather below informations how the continuity of ϕ is preserved when inducing F .
⇐ Additionally upper semicontinuity of a multifunction ϕ : X → K(X) with (pre)compact values is equivalent to upper Vietoris continuity of F .
Compact infinite system of (multi)functions, due to Theorem 1 with accompanying comments, can be turned into a system generated by a single multifunction and various continuity conditions are preserved during this process as shown in the following table: ⇒ The family of multifunctions ϕ i : X → P(X), i ∈ I, is assumed to be (pre)compact. In
Proof. For η/2 > 0 by continuity of ϕ with every x ∈ C we can associate The open cover U := {N λ(c) {c} : c ∈ C} of compact C admits by Lemma 1 a Lebesgue number λ > 0 i.e.