Dynamical Systems, Measures, and Fractals via Domain Theory

AbstractWe introduce domain theory in dynamical systems, iterated function systems (fractals), and measure theory. For a discrete dynamical system given by the action of a continuous map f: X → X on a metric space X, we study the extended dynamical systems (VX,Vf), (UX, Uf), and (LX, Lf), where V, U, and L are respectively the Vietoris hyperspace, the upper hyperspace, and the lower hyperspace functors. We show that if (X, f) is chaotic, then so is (UX, Uf). When X is locally compact UX, is a continuous bounded complete dcpo. If X is second countable as well, then UX will be ω-continuous and can be given an effective structure. We show how strange attractors, attractors of iterated function systems (fractals) and Julia sets are obtained effectively as fixed points of deterministic functions on UX or fixed points of non-deterministic functions on CUX where C is the convex (Plotkin) power domain. We also show that the set, M(X), of finite Borel measures on X can be embedded in PUX, where P is the probabilistic power domain. This provides an effective framework for measure theory. We then prove that the invariant measure of an hyperbolic iterated function system with probabilities can he obtained as the unique fixed point of an associated continuous function on PUX