When $K=[0,1]$ Weak Separation Condition coincides Finite type Condition

TL;DR

This paper investigates whether the Weak Separation Property (WSP) for certain similarity-based Iterated Function Systems (IFS) on the real line implies the Finite Type Condition, focusing on the case where [0,1] is the attractor.

Contribution

It examines the equivalence of WSP and Finite Type Condition for IFS with attractor [0,1], providing insights into their relationship in this specific setting.

Findings

WSP coincides with Finite Type Condition for the studied IFS case

The necessity of [0,1] as an attractor remains an open question

Explores conditions under which WSP implies finite type in simple IFS models

Abstract

This study is about the Iterated Function System (IFS) of similarities on $\mathbb R$ that satisfies Weak Separation property (WSP). We explore if this implies Finite type property. We look into the most simple case with condition that closed interval [0,1] is an attractor of the IFS. The necessity of this condition is not strict but still unknown.