Birkhoff's sphere conjecture for real-analytic area-preserving diffeomorphisms

Let $f$ be an orientation-preserving, real-analytic, Lebesgue measure-preserving diffeomorphism of the 2-sphere $\mathbb{S}^2$ with only two periodic points, which are necessarily fixed. Birkhoff's sphere conjecture. Then $f$ is conjugate to a rigid irrational rotation. This conjecture concerns the rigidity of area-preserving pseudo-rotations of the sphere and remains unsolved; it asks whether the absence of periodic orbits beyond the two fixed points forces analytic conjugacy to a rigid rotation.