Ergodicity of a Class of Cocycles Over Irrational Rotations

It is proved that if α is irrational and Φ̃ ∈ L 2 ( S 1 ) with Φ̃ ∈o(l/ n ) then for each m ∈ Z \{0} the corresponding skew product ( e 2 π ix , e 2 π iy ) ↦ ( e 2 π i ( x + α ) , e 2 π i ( Φ ( x ) + mx + y ) ) is ergodic. The rigidity of special flows over irrational rotations with roof functions whose Fourier coefficients are in o(l/ n ) is also shown.