We study a family of piecewise expanding maps on the plane, generated bycomposition of a rotation and an expansive similitude of expansion constant$\beta$. We give two constants $B_1$ and $B_2$ depending only on thefundamental domain that if $\beta>B_1$ then the expanding map has a uniqueabsolutely continuous invariant probability measure, and if $\beta>B_2$ then itis equivalent to $2$-dimensional Lebesgue measure. Restricting to a rotationgenerated by $q$-th root of unity $\zeta$ with all parameters in$\mathbb{Q}(\zeta,\beta)$, it gives a sofic system when $\cos(2\pi/q) \in\mathbb{Q}(\beta)$ and $\beta$ is a Pisot number. It is also shown that thecondition $\cos(2\pi/q) \in \mathbb{Q}(\beta)$ is necessary by giving a familyof non-sofic systems for $q=5$.
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