Rotational beta expansion: ergodicity and soficness

摘要

We study a family of piecewise expanding maps on the plane, generated by composition of a rotation and an expansive similitude of expansion constant β. We give two constants B_1 and B_2 depending only on the fundamental domain that if β ＞ B_1 then the expanding map has a unique absolutely continuous invariant probability measure, and if β ＞ B_2 then it is equivalent to 2-dimensional Lebesgue measure. Restricting to a rotation generated by q-th root of unity ζ with all parameters in Q(ζ, β), the map gives rise to a sofic system when cos(2π/q) ∈ Q(β) and β is a Pisot number. It is also shown that the condition cos(2π/q) ∈ Q(β) is necessary by giving a family of non-sofic systems for q = 5.