Texas Project NExT Journal
Vol. 1 (2003), pp. 1-9.

Non-Attractors of Iterated Function Systems

Manuel J. Sanders

MSC (2000): 37B25

Key words: arc, attractor, iterated function system, contraction, IFS, locally connected continuum

Abstract: An example of a locally connected continuum which is not the attractor of any iterated function system (IFS) in $\R ^2$ is constructed in a work of Kwieci\'{n}ski \cite{Kwiec}. The example is relative to a question of Hata \cite{Hata} regarding the existence of such continua. Kwieci\'{n}ski points out that a variation on his main construction provides an arc in $\R ^2$ which is not an attractor of any IFS. Here, criteria are developed that pertain to ascertaining whether or not a given arc embedded in Euclidean $n$-space can be realized as an attractor of some IFS. Besides reaffirming the example in \cite{Kwiec}, the characterization is sufficiently strong so as to provide many examples of arcs in $\R ^n (n \geq 2)$ which are not attractors of any IFS. Moreover, the techniques developed show that many arcs (as well as other compacta) in Euclidean spaces are indeed attractors of iterated function systems; techniques to construct such systems are developed naturally along the way. As a final note, there are arcs in $\R ^n$ that fail to meet the eligibility requirements for application of the developed criteria. Several open questions and conjectures regarding this issue will be discussed.

McMurry University, Abilene, TX 79697
e-mail: msanders@cs1.mcm.edu

Received by Editor: May 2, 2002.
Posted: June 9, 2003.

Copyright 2003