Continuity and chaos in discrete dynamical systems

Abstract

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.