A class of compact group extensions of β-transformations

We investigate the dynamical properties of a skew product transformation T φ on [ 0 , 1 ) × G defined by T φ ( x , g ) = ( T x , g ⋅ φ ( x ) ) where T is the β-transformation for β ⩾ 2 and φ ( x ) is a compact group G-valued step function with a finite number of discontinuities. We give several sufficient conditions for ergodicity and strong mixing of T φ . As an application, we describe a class of step functions which satisfy the Central Limit Theorem for the β-transformations. As another application, we also consider a class of skew product transformations T β , a , w on [ 0 , 1 ) × [ 0 , 1 ) which maps ( x , y ) ↦ ( β x , y + a x + w ) ( mod 1 ) where a , w ∈ R and give necessary and sufficient conditions for ergodicity and strong mixing.