[Paper Review] Non-hyperbolic Iterated Function Systems: attractors and stationary measures

Edgar Matias, Lorenzo J. Díaz|arXiv (Cornell University)|May 9, 2016

Mathematical Dynamics and Fractals18 references3 citations

TL;DR

This paper introduces a framework for analyzing non-hyperbolic iterated function systems (IFS) by focusing on weakly hyperbolic sequences—those for which the diameter of iterated images tends to zero. It establishes conditions under which such IFSs possess a unique strict attractor and characterizes the target set as the closure of the image under the coding map. The key contribution is a necessary and sufficient condition for the existence of a globally attracting fixed point of the Barnsley-Hutchinson operator, extending classical results beyond hyperbolic settings.

ABSTRACT

We consider iterated function systems $\mathrm{IFS}(T_1,\dots,T_k)$ consisting of continuous self maps of a compact metric space $X$. We introduce the subset $S_{\mathrm{t}}$ of {\emph{weakly hyperbolic sequences}} $ξ=ξ_0\ldotsξ_n \ldots \in Σ_k^+$ having the property that $\bigcap_n T_{ξ_{0}}\circ\cdots\circ T_{ξ_{n}}(X)$ is a point $\{π(ξ)\}$. The target set $π(S_{\mathrm{t}})$ plays a role similar to the semifractal introduced by Lasota-Myjak. Assuming that $S_{\mathrm{t}} e \emptyset$ (the only hyperbolic-like condition we assume) we prove that the IFS has at most one strict attractor and we state a sufficient condition guaranteeing that the strict attractor is the closure of the target set. Our approach applies to a large class of genuinely non-hyperbolic IFSs (e.g. with maps with expanding fixed points) and provides a necessary and sufficient condition for the existence of a globally attracting fixed point of the Barnsley-Hutchinson operator. We provide sufficient conditions under which the disjunctive chaos game yields the target set (even when it is not a strict attractor). We state a sufficient condition for the asymptotic stability of the Markov operator of a recurrent IFS. For IFSs defined on $[0,1]$ we give a simple condition for their asymptotic stability. In the particular case of IFSs with probabilities satisfying a "locally injectivity" condition, we prove that if the target set has at least two elements then the Markov operator is asymptotically stable and its stationary measure is supported in the closure of the target set.

Motivation & Objective

• To extend classical results on hyperbolic IFSs to genuinely non-hyperbolic systems with both contracting and expanding regions.
• To characterize the structure of attractors and stationary measures in IFSs lacking global uniform contraction.
• To identify conditions under which the Barnsley-Hutchinson operator has a unique global attractor despite non-hyperbolicity.
• To establish sufficient conditions for asymptotic stability of Markov operators and convergence of the chaos game to the target set.
• To clarify the relationship between the target set, its closure, and the existence of strict and Conley attractors in non-hyperbolic settings.

Proposed method

• Introduces the set $ S_{\mathrm{t}} \subset \Sigma_k^+ $ of weakly hyperbolic sequences $ \xi $ for which $ \mathrm{diam}(T_{\xi_0} \circ \cdots \circ T_{\xi_n}(X)) \to 0 $ as $ n \to \infty $.
• Defines the target set $ A_{\mathrm{t}} = \pi(S_{\mathrm{t}}) $ as the image of $ S_{\mathrm{t}} $ under the coding map $ \pi $, which maps sequences to points in $ X $.
• Uses the continuity of the Barnsley-Hutchinson operator $ \mathcal{B}_{\mathfrak{F}} $ to analyze convergence of iterates on compact sets and characterize attractors.
• Applies topological and measure-theoretic techniques to study asymptotic stability of Markov operators and stationary measures.
• Employs symbolic dynamics and residual set arguments to show that $ S_{\mathrm{t}} \neq \emptyset $ implies $ S_{\mathrm{t}} $ is residual in $ \Sigma_k^+ $, even without full weak hyperbolicity.
• Constructs explicit examples on $[0,1]$ to demonstrate cases where $ A_{\mathrm{t}} \subsetneq \overline{A_{\mathrm{t}}} $ or $ A_{\mathrm{t}} = \overline{A_{\mathrm{t}}} = [0,1] $, illustrating the role of local injectivity and contraction in compositions.

Experimental results

Research questions

• RQ1 Under what conditions does a non-hyperbolic IFS admit a unique strict attractor?
• RQ2 When is the target set $ A_{\mathrm{t}} $ equal to the closure of its image, and when is it a proper subset of its closure?
• RQ3 What conditions guarantee that the chaos game converges to the target set even when it is not a strict attractor?
• RQ4 When is the Markov operator of a recurrent IFS asymptotically stable?
• RQ5 What is the relationship between the support of the stationary measure and the closure of the target set in IFSs with locally injective maps?

Key findings

• If $ S_{\mathrm{t}} \neq \emptyset $, then the IFS has at most one strict attractor, and a sufficient condition ensures that the strict attractor is the closure of the target set $ A_{\mathrm{t}} $.
• The Barnsley-Hutchinson operator has a globally attracting fixed point if and only if $ S_{\mathrm{t}} \neq \emptyset $ and the target set is invariant under the IFS.
• For IFSs on $[0,1]$, asymptotic stability of the Markov operator holds under a simple condition on the maps' contraction and expansion behavior.
• When the IFS satisfies a 'locally injectivity' condition and $ A_{\mathrm{t}} $ has at least two points, the Markov operator is asymptotically stable and its stationary measure is supported in $ \overline{A_{\mathrm{t}}} $.
• In Example 6.2, $ A_{\mathrm{t}} $ is dense in $[0,1]$ but $ 1 \notin A_{\mathrm{t}} $, showing $ A_{\mathrm{t}} \subsetneq \overline{A_{\mathrm{t}}} $.
• In Example 6.3, $ A_{\mathrm{t}} = [0,1] $, demonstrating that even non-weakly hyperbolic IFSs can have full target set closure when certain compositions are uniformly contracting.