Birkhoff rigidity conjecture for analytic symplectic pseudo-rotations

Let $S$ be an orientable analytic surface. An analytic symplectomorphism of the sphere with only two fixed points and no other periodic point, or of a compact cylinder without periodic points, is considered a pseudo-rotation. Birkhoff's rigidity conjecture. Such a symplectomorphism is necessarily topologically conjugate to a rotation. The conjecture concerns rigidity of analytic area-preserving dynamics away from periodic points; the paper states that its constructions disprove it by producing analytic symplectomorphisms without periodic points that are not rigid.