Even unique intersection point can break OSC: an example

TL;DR

This paper provides a counterexample showing that a self-similar set with a unique intersection point among its pieces can still violate the open set condition, answering a long-standing question from the 1990s.

Contribution

It constructs a totally disconnected self-similar set with minimal overlaps that does not satisfy the open set condition, disproving a previously assumed implication.

Findings

Counterexample of a self-similar set with a unique intersection point

The set is totally disconnected and violates OSC

Minimal overlaps with only one intersection point

Abstract

This was a long-standing question since 90-ies whether one-point intersection property for a self-similar set implies open set condition. We answer this question negatively. We give an example of a totally disconnected self-similar set $K\subset \mathbb R$ which does not have open set condition and has minimal overlap of its pieces, that is, all intersections of its pieces $K_i\cap K_j,i\neq j$ are empty except only one, which is a single point.