Stability in the family of ω-limit sets of alternating systems

Let f and g be elements of C(I) with x∈I=[0,1]. We study the ω-limit sets ω(x,[f,g]) generated by alternating trajectories of the form γ(x,[f,g])={x,f(x),g(f(x)),f(g(f(x))),…}, as well as the sets Λ([f,g])=⋃x∈Iω(x,[f,g]) and L([f,g])={ω(x,[f,g]):x∈I}. In particular, we show that(1)If g is constant on no interval J⊆I, then there exists a residual set S⊆C(I) so that the maps Λ:C(I)×C(I)→K and L:C(I)×C(I)→K⋆ taking (f,g) to Λ([f,g]) and L([f,g]), respectively, are both continuous at (f,g) whenever f∈S.(2)The map ω:I×C(I)×C(I)→K taking (x,f,g) to ω(x,[f,g]) is in the second class of Baire, and for any g∈C(I) there exists a residual set T⊆I×C(I) so that ω is continuous at (x,f,g) whenever (x,f)∈T.(3)If f is constant on no interval J⊆I, then there exists a residual set D⊆I×C(I) so that ω(x,[f,g])=ω(x,g∘f)∪ω(f(x),f∘g), where both ω(x,g∘f) and ω(f(x),f∘g) are adding machines of type ∞, whenever (x,g)∈D.