A 1-dimensional Peano continuum which is not an IFS attractor

TL;DR

This paper constructs a specific 1-dimensional Peano continuum that cannot be realized as an attractor of an iterated function system, answering a longstanding open question.

Contribution

It provides the first known example of a 1-dimensional Peano continuum that is not an IFS attractor, resolving an open problem in the field.

Findings

Constructed a 1-dimensional Peano continuum not homeomorphic to any IFS attractor.

Answered a long-standing question posed by M. Hata.

Demonstrated limitations of IFS attractors in representing certain continua.

Abstract

Answering an old question of M.Hata, we construct an example of a 1-dimensional Peano continuum which is not homeomorphic to an attractor of IFS.