Dynamical systems, Measures and Fractals via Domain Theory

Abstract

We introduce domain theory in the computation of dynamical systems, iterated function systems (fractals) and measures. For a discrete dynamical system (X, f), given by the action of a continuous map f: X → X on a metric space X, we study the extended dynamical systems (VX, Vf) and (UX, Uf) where V is the Vietoris functor and U is the upper space functor. In fact, from the point of view of computing the attractors of (X, f), it is natural to study the other two systems: A compact attractor of (X, f) is a fixed point of (VX, Vf) and a fixed point of (UX, Uf). 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 establish an interesting link between measure theory and domain theory. We show that the set, M(X), of Borel measures on X can be embedded in PUX, where P is the probabilistic power domain. This provides an effective way of obtaining measures on X. We then prove that the invariant measure of an hyperbolic iterated function system with probabilities can be obtained as the unique fixed point of an associated continuous function on PUX.