Open Access
January, 2017 Rotational beta expansion: ergodicity and soficness
Shigeki AKIYAMA, Jonathan CAALIM
J. Math. Soc. Japan 69(1): 397-415 (January, 2017). DOI: 10.2969/jmsj/06910397


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 $\mathbb{Q}(\zeta,\beta)$, the map gives rise to 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. "Rotational beta expansion: ergodicity and soficness." J. Math. Soc. Japan 69 (1) 397 - 415, January, 2017.


Published: January, 2017
First available in Project Euclid: 18 January 2017

zbMATH: 1379.37010
MathSciNet: MR3597559
Digital Object Identifier: 10.2969/jmsj/06910397

Primary: 37A45
Secondary: 11K16 , 37B10 , 37E05

Keywords: beta expansion , invariant measure , Pisot number , sofic system

Rights: Copyright © 2017 Mathematical Society of Japan

Vol.69 • No. 1 • January, 2017
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