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