Graph Directed Markov Systems on Hilbert Spaces

We deal with contracting finite and countably infinite iterated function systems acting on Polish spaces, and we introduce conformal Graph Directed Markov Systems on Polish spaces. Sufficient conditions are provided for the closure of limit sets to be compact, connected, or locally connected. Conformal measures, topological pressure, and Bowen’s formula (determining the Hausdorff dimension of limit sets in dynamical terms) are introduced and established. We show that, unlike the Euclidean case, the Hausdorff measure of the limit set of a finite iterated function system may vanish. Investigating this issue in greater detail, we introduce the concept of geometrically perfect measures and we proved sufficient conditions. Geometrical perfectness guarantees the Hausdorff measure of the limit set to be primitive. As a by–product of the mainstream of our investigations we prove 4r–covering theorem for all metric spaces. It enables us to establish appropriate co–Frostman type theorems.