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According to the Denjoy theorem, every orientation-preserving C2-diffeomorphism of the circle with an irrational rotation number θ is topologically conjugate to Tθ.
As a consequence of the Denjoy theorem, an orientation preserving C2 diffeomorphism ƒ of the circle is structurally stable if and only if its rotation number is rational, ρ(ƒ) = p/q, and the periodic trajectories, which all have period q, are non-degenerate: the Jacobian of ƒq at the periodic points is different from 1, cf Circle map.