Summary.
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.
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Manuscript received: September 9, 2004 and, in final form, April 4, 2005.
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Steele, T.H. Continuity and chaos in discrete dynamical systems. Aequ. math. 71, 300–310 (2006). https://doi.org/10.1007/s00010-005-2813-7
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DOI: https://doi.org/10.1007/s00010-005-2813-7