Real Projective Iterated Function Systems

TL;DR

This paper explores the properties of attractors in projective iterated function systems, providing characterizations, uniqueness results, duality connections, and an invariant related to hyperplane-avoiding attractors.

Contribution

It introduces new theorems characterizing hyperplane-avoiding attractors, proves the uniqueness of attractors, and defines a projective invariant for such attractors.

Findings

Characterization of hyperplane-avoiding attractors

Proof of at most one attractor for a projective IFS

Definition of a nontrivial projective invariant

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.