A Positive Operator-Valued Measure for an Iterated Function System

Given an iterated function system (IFS) on a complete and separable metric space Y, there exists a unique compact subset X. Y satisfying a fixed point relation with respect to the IFS. This subset is called the attractor set, or fractal set, associated to the IFS. The attractor set supports a specific Borel probability measure, called the Hutchinson measure, which itself satisfies a fixed point relation. P. Jorgensen generalized the Hutchinson measure to a projection-valued measure, under the assumption that the IFS does not have essential overlap (Jorgensen in Adv. Appl. Math. 34(3): 561-590, 2005; Operator Theory, Operator Algebras, and Applications, pp. 13-26, 2006). In previous work, we developed an alternative approach to proving the existence of this projection-valued measure (Davison in Acta Appl. Math. 140(1): 11-22, 2015; Acta Appl. Math. 140(1): 23-25, 2015; Generalizing the Kantorovich metric to projection-valued measures: with an application to iterated function systems, 2015). The situation when the IFS exhibits essential overlap has been studied by Jorgensen and colleagues in Jorgenson et al. (J. Math. Phys. 48(8): 083511, 35, 2007). We build off their work to generalize the Hutchinson measure to a positive-operator valued measure for an arbitrary IFS, that may exhibit essential overlap. This work hinges on using a generalized Kantorovich metric to define a distance between positive operator-valued measures. It is noteworthy to mention that this generalized metric, which we use in our previous work as well, was also introduced by R. F. Werner to study the position and momentum observables, which are central objects of study in the area of quantum theory (Werner in J. Quantum Inf. Comput. 4(6): 546-562, 2004). We conclude with a discussion of Naimark's dilation theorem with respect to this positive operator-valued measure, and at the beginning of the paper, we prove a metric space completion result regarding the classical Kantorovich metric.