[Paper Review] The chaos game on a general iterated function system from a topological point of view

Michael F. Barnsley, Krzysztof Leśniak|arXiv (Cornell University)|Mar 2, 2012

Mathematical Dynamics and Fractals23 references4 citations

TL;DR

This paper establishes that, for a general iterated function system (IFS) on a complete metric space, the chaos game orbit generated by a disjunctive sequence converges to the attractor's full fibre structure, except on a σ-porous set of initial points. The key contribution is a topological Baire category result showing that almost all orbits (in the category sense) yield the attractor, even when the IFS is non-contractive and has multiple attractors.

ABSTRACT

We investigate combinatorial issues relating to the use of random orbit approximations to the attractor of an iterated function system with the aim of clarifying the role of the stochastic process during generation the orbit. A Baire category counterpart of almost sure convergence is presented; and a link between topological and probabilistic methods is observed.

Motivation & Objective

• To investigate the convergence of chaos game orbits in general iterated function systems (IFS) without requiring contractivity.
• To establish a topological counterpart to almost sure convergence in stochastic chaos games, using Baire category theory.
• To clarify the role of the underlying sequence (deterministic or stochastic) in generating orbits that intersect all fibres of an attractor.
• To demonstrate that disjunctive sequences—common in data analysis—ensure robust convergence to attractors even in non-standard IFS settings.
• To introduce and prove the 'Rapunzel Theorem', showing that orbits escape repellers under disjunctive sequences, even when starting in the dual repeller's interior.

Proposed method

• Define attractors and basins of attraction for general continuous IFSs on complete metric spaces, without assuming contractivity.
• Introduce the concept of fibres of an attractor and classify attractors as minimal-fibred, strongly-fibred, or point-fibred based on fibre structure.
• Use disjunctive sequences—sequences containing every finite word over the alphabet Σ—as a topological substitute for random sequences in the chaos game.
• Prove that the tail of any disjunctive chaos game orbit converges in the Hausdorff metric to a set that intersects every fibre of the attractor.
• Apply Baire category theory to show that the set of initial points for which the chaos game fails to converge to a strongly-fibred attractor is σ-porous, a stronger notion than first category.
• Establish the 'Rapunzel Theorem' by showing that even if the initial point lies in the dual repeller (complement of the basin), a disjunctive sequence ensures escape and convergence to the attractor.

Experimental results

Research questions

• RQ1 Under what topological conditions does the chaos game orbit converge to the full attractor structure, even when the IFS is not contractive?
• RQ2 How does the use of disjunctive sequences in the chaos game compare to stochastic processes in terms of convergence to attractors?
• RQ3 Can a topological notion like Baire category provide a stronger or more general convergence guarantee than probabilistic almost-sure convergence?
• RQ4 What happens to the chaos game when the initial point lies in the dual repeller of a point-fibred attractor/repeller pair?
• RQ5 Can the chaos game on a deterministic data string (e.g., DNA sequence) be guaranteed to reveal the underlying attractor structure via topological means?

Key findings

• The tail of any disjunctive chaos game orbit, starting from any point in the basin of an attractor, converges in the Hausdorff metric to a set that intersects every fibre of the attractor.
• For a strongly-fibred attractor, the set of initial points for which the chaos game fails to converge to the attractor is σ-porous, implying it is small in the Baire category sense.
• A disjunctive stochastic process generates chaos game orbits that converge to a strongly-fibred attractor with probability one.
• The 'Rapunzel Theorem' proves that even if the initial point lies in the dual repeller (which has nonempty interior), a disjunctive sequence ensures the orbit escapes and converges to the attractor.
• The complement of the set of points yielding the attractor under a disjunctive sequence is σ-porous, which is stronger than first category, indicating robustness of convergence.
• The results generalize classical chaos game convergence beyond contractive IFSs, showing that combinatorial and topological properties (like disjunctiveness) can replace stochastic assumptions.