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 $β$. 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 $\mathbb{Q}(ζ,β)$, it gives a sofic system when $\cos(2π/q) \in \mathbb{Q}(β)$ and $β$ is a Pisot number. It is also shown that the condition $\cos(2π/q) \in \mathbb{Q}(β)$ is necessary by giving a family of non-sofic systems for $q=5$.

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