Shift limits of a non-autonomous system

Let t=t1t2⋯ be an element of the full shift with map τ on a finite alphabet symbols A and let Σ=closure{τi(t):i∈N∪{0}}. ft=ft1,∞=⋯∘ft2∘ft1 non-autonomous system over compact metric space X where ti∈A. The set Ft+={fτi(t):i∈N} is called shifted family ft. If t transitive in AN, then Ft+‾ classical IFS; otherwise, Σ subshift Ft+‾={fσ=fσ1,∞:σ∈Σ} “generalized” IFS. We show that if ft has some shadowing specification properties, fσ∈Ft+‾ those properties as well; however, this claim not true for other such transitivity, mixing exactness. Also, sofic x∈X periodic fσ∈Ft+‾, σ necessarily periodic, there σ′∈Σ x fσ′∈Ft+‾.