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arXiv · 1502.01793

Rotational beta expansion: Ergodicity and Soficness

Shigeki Akiyama·Jonathan Caalim
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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 $q=5$.

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Shigeki Akiyama, Jonathan Caalim. 2015-02-06. Rotational beta expansion: Ergodicity and Soficness. https://arxiv.org/abs/1502.01793

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