Fractals and domain theory

Abstract

We show that a measurement $\mu$ on a continuous dcpo $D$ extends to a measurement $\skew3\bar{\mu}$ on the convex powerdomain ${\mathbf C} D$ iff it is a Lebesgue measurement. In particular, $\ker\mu$ must be metrisable in its relative Scott topology. Moreover, the space $\ker\skew3\bar{\mu}$ in its relative Scott topology is homeomorphic to the Vietoris hyperspace of $\ker\mu$, that is, the space of non-empty compact subsets of $\ker\mu$ in its Vietoris topology – the topology induced by any Hausdorff metric. This enables one to show that Hutchinson's theorem holds for any finite set of contractions on a domain with a Lebesgue measurement. Finally, after resolving the existence question for Lebesgue measurements on countably based domains, we uncover the following relationship between classical analysis and domain theory: for an $\omega$-continuous dcpo $D$ with $\max(D)$ regular, the Vietoris hyperspace of $\max(D)$ embeds in $\max({\mathbf C} D)$ as the kernel of a measurement on ${\mathbf C} D$.