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 $\beta$. 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 $\mathbb{Q}(\zeta ,\beta )$, 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 the condition $\cos (2\pi /q)\in \mathbb{Q}(\beta )$ is necessary by giving a family of non-sofic systems for $q=5$.

Revised version: to appear in JMSJ after certain edition