A positive answer to a symmetry conjecture on homogeneous IFS

TL;DR

This paper proves that if two homogeneous iterated function systems (IFS) with opposite contraction factors share the same attractor on the real line and satisfy the open set condition, then the attractor must be symmetric, answering a conjecture.

Contribution

It provides a positive resolution to a symmetry conjecture for homogeneous IFSs with opposite contraction factors sharing the same attractor.

Findings

Shared attractor is symmetric under given conditions

Confirms a conjecture posed by Feng and Wang

Advances understanding of symmetry in IFS attractors

Abstract

If two homogeneous IFSs satisfying the OSC with opposite common contraction factors share the same attractor on the real line, we show that this attractor is symmetric. This answers a question of Feng and Wang [Adv. Math. 222 (2009), 1964-1981].