Rotational beta expansion: ergodicity and soficness

We study a family of piecewise expanding maps on the plane, generated by composition of a rotation and an expansive similitude of expansion constant beta. We give two constants B-1 and B-2 depending only on the fundamental domain that if beta > B-1 then the expanding map has a unique absolutely continuous invariant probability measure, and if beta > B-2 then it is equivalent to 2-dimensional Lebesgue measure. Restricting to a rotation generated by q-th root of unity zeta with all parameters in Q(zeta,beta), the map gives rise to a sofic system when cos(2 pi/q) is an element of Q(beta) and beta is a Pisot number. It is also shown that the condition cos(2 pi/q) is an element of Q(beta) is necessary by giving a family of non-sofic systems for q = 5.