On cut sets of attractors of iterated function systems

TL;DR

This paper investigates the structure of cut sets in attractors of iterated function systems, providing criteria for cut points and methods to identify them, especially in self-affine tiles, with algorithmic implications.

Contribution

It establishes that irreducible cut sets are either perfect sets or points and offers an algorithmic approach to detect cut points in self-affine tiles.

Findings

Irreducible cut sets are either perfect sets or single points.

Provides a criterion for the existence of cut points in IFS attractors.

Develops an algorithmic method using Hata graphs to find cut points.

Abstract

In this paper, we study cut sets of attractors of iteration function systems (IFS) in $\mathbb{R}^d$. Under natural conditions, we show that all irreducible cut sets of these attractors are perfect sets or single points. This leads to a criterion for the existence of cut points of IFS attractors. If the IFS attractors are self-affine tiles, our results become algorithmically checkable and can be used to exhibit cut points with the help of Hata graphs. This enables us to construct cut points of some self-affine tiles studied in the literature.