Smoothness of Invariant Curves

If F is a C^3+γ area-preserving surface diffeomorphism with an invariant curve that is a C^2+β topological circle https://www.w3.org/1998/Math/MathML">Chttps://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332838/58f4c041-1317-4a7c-b51a-e794ca13209e/content/eq2165.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> on which F acts like a rotation by a “typical” irrational rotation number, then https://www.w3.org/1998/Math/MathML">Chttps://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332838/58f4c041-1317-4a7c-b51a-e794ca13209e/content/eq2166.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> is in fact C^2+γ′ for all γ′ < γ.