Dynamical Systems, Measures and Fractals Via Domain Theory

We 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 (V X; V f), (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 !-continuous and can be given an eeective structure. We show how strange attractors, attractors of iterated function systems (fractals) and Julia sets are obtained eeectively as xed points of deterministic functions on UX or xed points of non-deterministic functions on CUX where C is the convex (Plotkin) power domain. We also show that the set, M(X), of nite Borel measures on X can be embedded in P UX, where P is the probabilistic power domain. This provides an eeective framework for measure theory. We then prove that the invariant measure of an hyperbolic iterated function system with probabilities can be obtained as the unique xed point of an associated continuous function on P UX.