Heteroclinic orbits and chaotic invariant sets for monotone twist maps

We consider the monotone twist map f̄ on (ℝ/ℤ) × ℝ, its lift f on ℝ² and its associated variational principle h: ℝ² → through its generating function. By working with the variational principle h, we first show that for an adjacent minimal chain {(u^k, v^k)}_k=s^t of fixed points of f, if there exists a barrier B_k for each adjacent minimal pair u^k < u^k+1, s ≤ k ≤ t - 1, then there exists a heteroclinic orbit between (u^s, v^s) and (u^t, v^t), then by assuming that there is a barrier for any two neighboring globally minimal critical points and m is sufficiently large, we construct an invariant set Λ^m ⊂ (ℝ/ℤ) × ℝ such that the shift map of the n-symbol space is a factor of f̄^mΛ^m, where n is the total number of the globally minimal fixed points of f̄.