Invariant Graph: Tθ(ϕ(θ))=ϕ(θ +ω)

Question

Let M= S1 × [a, b],T : M → M, (θ , x)→ (θ + ω, Tθ (x)). Also let ∀θ ∈ S1 ,Tθ : [a, b] → [a, b] is increasing and c.t.s. in x,θ s.t. Tθ(a)> a, Tθ(b)< b and K ⊂ M is a non empty compact set with T (K) = K. Define ϕ(θ)= sup{x ∈ [a, b] : (θ , x) ∈ K}.Show that the graph of ϕ is invariant under T i.e. Tθ(ϕ(θ))=ϕ(θ +ω).Where ω is a fixed irrational number and S1 is unit circle and θ + ω is calculated with mod 1.

Answer 1

I think since Tθ : [a, b] → [a, b] is increasing and c.t.s. it preserves the supremum.But I can not complete the proof.