Journal of the Mathematical Society of Japan

Rotational beta expansion: ergodicity and soficness

Shigeki AKIYAMA and Jonathan CAALIM

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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 $q=5$.

Article information

J. Math. Soc. Japan, Volume 69, Number 1 (2017), 397-415.

First available in Project Euclid: 18 January 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37A45: Relations with number theory and harmonic analysis [See also 11Kxx]
Secondary: 37B10: Symbolic dynamics [See also 37Cxx, 37Dxx] 37E05: Maps of the interval (piecewise continuous, continuous, smooth) 11K16: Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc. [See also 11A63]

beta expansion invariant measure sofic system Pisot number


AKIYAMA, Shigeki; CAALIM, Jonathan. Rotational beta expansion: ergodicity and soficness. J. Math. Soc. Japan 69 (2017), no. 1, 397--415. doi:10.2969/jmsj/06910397.

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