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

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- by J. C. Perry PDF
- Proc. Amer. Math. Soc.
**124**(1996), 2479-2489 Request permission

## 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 $l^2(A)$.## References

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*The Fractal Geometry of Nature*, W. H. Freeman and Company, New York, 1983. - Michael Barnsley,
*Fractals everywhere*, Academic Press, Inc., Boston, MA, 1988. MR**977274** - Uroš Milutinović,
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## Additional Information

**J. C. Perry**- Affiliation: Systems Research and Technology Department, Naval Surface Warfare Center, Dahlgren, Virginia 22448
- 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
- © Copyright 1996 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**124**(1996), 2479-2489 - MSC (1991): Primary 51F99, 54C25, 54F45
- DOI: https://doi.org/10.1090/S0002-9939-96-03554-X
- MathSciNet review: 1346984