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 ∈ I ω ( x , [ f , g ] ) and L ( [ f , g ] ) = { ω ( x , [ f , g ] ) : x ∈ I } . In particular, we show that(1) s 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 . p 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 . s 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 .