[Paper Review] On the continuity of the Hutchinson operator

Michael F. Barnsley, Krzysztof Leśniak|arXiv (Cornell University)|Feb 12, 2012

Mathematical Dynamics and Fractals9 references4 citations

TL;DR

This paper establishes the continuity of the Hutchinson operator induced by an iterated function system (IFS) when the underlying maps are continuous, particularly proving that continuity of the component functions implies continuity of the operator on the hyperspace of compact sets under the Hausdorff metric. The key contribution is a rigorous proof using uniform continuity on compact sets and Lebesgue number arguments, resolving long-standing questions about the stability of attractors under perturbations of non-contractive maps.

ABSTRACT

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.

Motivation & Objective

• To resolve open questions about the continuity of the Hutchinson operator when component maps are not necessarily contractive.
• To establish conditions under which the Hutchinson operator remains continuous despite the absence of strict contractivity.
• To clarify the relationship between continuity of individual maps and continuity of the induced operator on the hyperspace of compact sets.
• To provide a theoretical foundation for the stability of attractors in IFS with non-contractive, projective-type maps.

Proposed method

• Uses the Hausdorff metric on the hyperspace of nonempty compact subsets of a complete metric space to analyze operator continuity.
• Applies the Cantor–Weierstrass uniform continuity theorem to continuous multifunctions on compact sets, leveraging Lebesgue number properties.
• Establishes a chain of implications: continuity of component maps → uniform continuity on compact sets → continuity of the induced Hutchinson operator.
• Employs the inequality $ h(ar{ ho}(N_ ho B), ar{ ho}(B)) o 0 $ as $ ho \to 0 $, ensuring the operator preserves proximity in the Hausdorff topology.
• Uses the inclusion $ F(N_ ho B) \subset N_\varepsilon F(B) $ for small $ \rho $, derived from uniform continuity, to prove continuity at compact sets.
• Applies the Lebesgue number lemma to construct a uniform $ \delta $-neighborhood such that $ h(B,C) < \delta \Rightarrow h(F(B),F(C)) < \varepsilon $.

Experimental results

Research questions

• RQ1 Under what conditions is the Hutchinson operator continuous when the underlying maps are not contractive?
• RQ2 Does continuity of the component functions in an IFS imply continuity of the induced Hutchinson operator on the hyperspace of compact sets?
• RQ3 Can uniform continuity on compact sets be used to establish continuity of the Hutchinson operator in the Hausdorff metric?
• RQ4 How does the structure of the hyperspace $ \mathcal{K}(X) $ with the Hausdorff metric affect the continuity of the operator?

Key findings

• The Hutchinson operator $ F: \mathcal{K}(X) \to \mathcal{K}(X) $ is continuous at every compact set $ C \in \mathcal{K}(X) $ if the component multifunctions are continuous.
• Continuity of the individual maps $ f_i $ in a finite or compact IFS implies continuity of the induced Hutchinson operator on the hyperspace of compacta.
• The proof relies on uniform continuity of the multifunction on compact sets and the existence of a Lebesgue number for open covers, ensuring uniform control over $ h(F(B), F(C)) $.
• A necessary condition is established: if $ F $ is continuous at singletons, then the underlying multifunction $ \varphi $ must be continuous.
• The result holds even when the maps are not contractive, extending the theory beyond hyperbolic IFSs to include projective and non-contractive systems.
• The operator is continuous at $ C \in \mathcal{K}(X) $ if for every $ \varepsilon > 0 $, there exists $ \delta > 0 $ such that $ h(B,C) < \delta \Rightarrow h(F(B),F(C)) < \varepsilon $, proven via neighborhood inclusions and Hausdorff distance estimates.