Abstract For a surface diffeomorphism f ∈Diff l ( M ), with l ≥8, we prove that if f exhibits a non-transversal heteroclinic cycle composed of two fixed saddle points Q 1 and Q 2 , one dissipative and the other expansive, then there exists an open set 𝒱⊂Diff l ( M ) such that $ f \in \overline {\mathcal {V}}$ and there exists a dense set 𝒟⊂𝒱 such that for all g ∈𝒟, g exhibits infinitely many invariant periodic curves with irrational rotation numbers. Moreover, these curves are C 1 conjugated to an irrational rotation on 𝕊 1 .
