According to the Denjoy theorem, every orientation-preserving C2-diffeomorphism of the circle with an irrational rotation number θ is topologically conjugate to Tθ.
In mathematics, the Radó–Kneser–Choquet theorem, named after Tibor Radó, Hellmuth Kneser and Gustave Choquet, states that the Poisson integral of a homeomorphism of the unit circle is a harmonic diffeomorphism of the open unit disk.
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.
In mathematics, a Veech surface is a translation surface (X,ω) (a Riemann surface X with a holomorphic 1-form ω) whose group SL(X,ω) of affine diffeomorphisms is a lattice in SL2(R) (a discrete subgroup of cofinite volume).