A Topological Separation Condition for Fractal Attractors

TL;DR

This paper introduces a new topological separation condition for fractal attractors, showing its equivalence to the strong Markov property and analyzing its prevalence among contractive homeomorphisms.

Contribution

It defines a separation condition weaker than the strong open set condition, proves its equivalence to the strong Markov property, and establishes density results for non-redundant contractive systems.

Findings

Separation condition is equivalent to the strong Markov property.

The set of non-redundant N-tuples is a G_delta set.

Density results for fixed points of contraction matrices.

Abstract

We consider finite systems of contractive homeomorphisms of a complete metric space, which are non-redundant on every level. In general this separation condition is weaker than the strong open set condition and is not equivalent to the weak separation property. We prove that this separation condition is equivalent to the strong Markov property (see definition below). We also show that the set of $N$-tuples of contractive homeomorphisms, which are non-redundant on every level, is a $G_\delta$ set in the topology of pointwise convergence of every component mapping with an additional requirement that the supremum of contraction coefficients of mappings be strictly less than one. We give several sufficient conditions for this separation property. For every fixed $N$-tuple of $d\times d$ invertible contraction matrices from a certain class, we obtain density results for $N$-tuples of fixed points which define $N$-tuples of mappings non-redundant on every level.