Rotational beta expansion: ergodicity and soficness

Abstract

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 B₁ and B₂ depending only on the fundamental domain that if β > B₁ then the expanding map has a unique absolutely continuous invariant probability measure, and if β > B₂ 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.