Given an iterated function system (IFS) on a complete and separable metric space Y, there exists a unique compact subset X contained in Y satisfying a fixed point relation with respect to this IFS. This subset is called the attractor set, or fractal set, associated to this IFS. The attractor set supports a specific Borel probability measure, called the Hutchinson measure, which itself satisfies a fixed point relation. A particular instance of an attractor set is the infinite string space on a finite alphabet, with the members of the IFS being the shift maps. There is known relationship between the Hutchinson measure on the infinite string space, and the Hutchinson measure on an arbitrary attractor set, via the so called coding map. In this talk, we will generalize this relationship to the functional analytic setting, building off the work of Palle Jorgensen (U of Iowa). This generalized relationship will be an example of Naimark's dilation theorem for positive operator-valued measures.
Operator-Valued Measures Associated to Iterated Function Systems