Continuity and chaos in discrete dynamical systems

We study the map ω: I × C(I I) → K given by (x f) → ω (x f) that takes a point x in the unit interval I = [0 1] and f a continuous self-map of the unit interval to the ω-limit set ω (x f) that together they generate. We characterize those points (x f) I × C(I I) at which ω : I × C(I I) → K is continuous and show that ω: I × C(I I) → K is in the second class of Baire. We also consider the trajectory map τ : I × C(I I) → l ∞ given by (x f) → τ (x f) = {x f(x) f(f(x)) ... } and find that both ω: I × C(I I) → K and τ : I × C(I I) → l ∞ are continuous on a residual subset of I × C(I I). We show that the Hausdorff s-dimensional measure of an ω-limit set is typically zero for every s > 0. © Birkhäuser Verlag, Basel, 2006.