LAUSR

Abstract Let I = [ 0 , 1 ] with b B 1 the set of Baire-1 self-maps of I. There exists S a residual subset of b B 1 such that for any f ∈ S , the following hold: 1. The set of the ω-limit points Λ ( f ) = ∪ x ∈ I ω ( x , f ) is a nowhere dense and perfect subset of [ 0 , 1 ] with Hausdorff dimension zero. 2. The collection of the ω-limit sets Ω ( f ) = { ω ( x , f ) : x ∈ I } generated by f is closed in the Hausdorff metric space. 3. If x is a point at which f is continuous, then ( x , f ) is a point at which the map ω : I × b B 1 → K given by ( x , f ) → ω ( x , f ) is continuous. 4. The n-fold iterate f n is an element of b B 1 for all natural numbers n.