On weak separation property for self-affine Jordan arcs

TL;DR

This paper investigates the weak separation property of self-affine Jordan arcs in the plane, establishing that failure of this property characterizes parabolic segments, and otherwise such arcs are attractors of multizippers.

Contribution

It proves that non-weakly separated self-affine Jordan arcs are parabolic segments, and if not, they are attractors of multizippers, clarifying their geometric structure.

Findings

Violation of weak separation implies the arc is a parabolic segment.

If not a parabolic segment, the arc is an attractor of some multizipper.

Provides a characterization of self-affine Jordan arcs based on separation properties.

Abstract

We consider self-affine arcs in $\mathbb R^2$ and prove that violation of "inner" weak separation property for such arcs implies that the arc is a parabolic segment. Therefore, if a self-affine Jordan arc is not a parabolic segment, then it is the attractor of some multizipper.