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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Lipscomb's universal space is the attractor
of an infinite iterated function system


Author: J. C. Perry
Journal: Proc. Amer. Math. Soc. 124 (1996), 2479-2489
MSC (1991): Primary 51F99, 54C25, 54F45
MathSciNet review: 1346984
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Abstract: Lipscomb's one-dimensional space $L(A)$ on an arbitrary index set $A$ is injected into the Tychonoff cube $I^A$. The image of $L(A)$ is shown to be the attractor of an iterated function system indexed by $A$. This system is conjugate, under an injection, with a set of right-shift operators on Baire's space $N(A)$ regarded as a code space. This view of $L(A)$ extends the fractal nature of $L(A)$ initiated in a 1992 joint paper by the author and S. Lipscomb. In addition, we give a new proof that as a subspace of Hilbert's space $l^2(A)$, the space $L(A)$ is complete and hence is closed in $l^2(A)$.


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Additional Information

J. C. Perry
Affiliation: Systems Research and Technology Department, Naval Surface Warfare Center, Dahlgren, Virginia 22448

DOI: http://dx.doi.org/10.1090/S0002-9939-96-03554-X
PII: S 0002-9939(96)03554-X
Keywords: Dimension theory, Lipscomb's space, fractals, infinite iterated function system
Received by editor(s): October 10, 1993
Additional Notes: This work was partially supported by research grants from the Naval Surface Warfare Center.
Communicated by: James E. West
Article copyright: © Copyright 1996 American Mathematical Society