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

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 $l^2(A)$.