A class of compact group extensions of β-transformations

We investigate the dynamical properties of a skew product transformation Tφ on [0,1)×G defined by Tφ(x,g)=(Tx,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 Tφ. 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 Tβ,a,w on [0,1)×[0,1) which maps where a,w∈R and give necessary and sufficient conditions for ergodicity and strong mixing.