Ergodicity and weak-mixing of homogeneous extensions of measure-preserving transformations with applications to Markov shifts

Given a homogeneous extension S' of a measure-preserving transformation T, we provide necessary and sufficient conditions for the ergodicity and weak-mixing of S' in terms of functional equations. We then apply our findings to the case when Tis a Markov shift and the associated skewing function of S' depends on a finite number of coordinates. In this case, we obtain a simplification to the appropriate functional equations.