[Paper Review] Real Projective Iterated Function Systems

Michael F. Barnsley, Andrew Vince|arXiv (Cornell University)|Mar 17, 2010

Mathematical Dynamics and Fractals20 references4 citations

TL;DR

This paper establishes foundational theorems for real projective iterated function systems (IFS), proving that a projective IFS has at most one attractor and characterizing when such an attractor avoids a hyperplane. It introduces a novel, nontrivial projective invariant—'index'—which is preserved under projective transformations and distinguishes attractors in classical projective geometry.

ABSTRACT

This paper contains four main results associated with an attractor of a projective iterated function system (IFS). The first theorem characterizes when a projective IFS has an attractor which avoids a hyperplane. The second theorem establishes that a projective IFS has at most one attractor. In the third theorem the classical duality between points and hyperplanes in projective space leads to connections between attractors that avoid hyperplanes and repellers that avoid points as well as hyperplane attractors that avoid points and repellers that avoid hyperplanes. Finally, an index is defined for attractors which avoid a hyperplane. This index is shown to be a nontrivial projective invariant.

Motivation & Objective

• To establish rigorous existence and uniqueness conditions for attractors in real projective iterated function systems (IFS).
• To characterize when an attractor avoids a hyperplane, linking this to contractivity and duality in projective geometry.
• To define and analyze a new projective invariant—'index'—for attractors that avoid a hyperplane.
• To explore the duality between attractors avoiding hyperplanes and repellers avoiding points, revealing deeper geometric structure.
• To demonstrate that the index is a nontrivial projective invariant, joining cross ratio and Hausdorff dimension in geometric invariance theory.

Proposed method

• Uses the equivalence of five conditions (Theorem 1) to characterize when a projective IFS has an attractor avoiding a hyperplane, including contractivity on a compact set and existence of nested convex bodies.
• Applies projective duality to relate attractors avoiding hyperplanes with repellers avoiding points, and vice versa, using the adjoint IFS $\mathcal{F}^t$.
• Defines the 'index' of an attractor as the number of times a certain composition of maps wraps around a convex body, using a recursive construction of a new IFS $\mathcal{G}$ from $\mathcal{F}$.
• Proves that the index is invariant under projective transformations by showing $\text{index}(\mathcal{G}) = \text{index}(\mathcal{F})$ via a construction involving compositions of maps.
• Employs the Hilbert metric $d_K$ on convex bodies to establish bi-Lipschitz equivalence and invariance of Hausdorff dimension under projective maps.
• Leverages results from spectral theory and the Krein-Rutman theorem to support the existence of invariant structures in the IFS dynamics.

Experimental results

Research questions

• RQ1 Under what conditions does a projective IFS have an attractor that avoids a hyperplane?
• RQ2 Can the uniqueness of an attractor in a projective IFS be established independently of contractivity on a metric space?
• RQ3 How does projective duality between points and hyperplanes manifest in the dynamics of IFS attractors and repellers?
• RQ4 What is the nature of the new invariant—'index'—and how does it behave under projective transformations?
• RQ5 Is the index a nontrivial projective invariant, and how does it compare to other invariants like cross ratio and Hausdorff dimension?

Key findings

• A projective IFS has at most one attractor, establishing a fundamental uniqueness property in the projective setting.
• An attractor avoids a hyperplane if and only if the IFS is contractive on a nonempty open set avoiding a hyperplane, as formalized in Theorem 1’s five equivalent conditions.
• The index of an attractor that avoids a hyperplane is a nontrivial projective invariant, with the paper proving $\text{index}(A) = 2$ for a specific example involving a circle arc.
• The index is preserved under projective transformations, as shown by constructing a new IFS $\mathcal{G}$ with the same index as the original $\mathcal{F}$.
• The attractor's index is independent of the choice of convex body and is defined via a recursive composition of maps that eventually map a set into a proper subset.
• The Hausdorff dimension of an attractor is invariant under projective transformations, as it is preserved under bi-Lipschitz equivalence of the Hilbert and round metrics.