Year: 2015
Type: preprint
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 $β$. We give two constants $B_1$ and $B_2$ depending only on the fundamental domain that if $β>B_1$ then the expanding map has a unique absolutely continuous invariant probability measure, and if $β>B_2$ then it is equivalent to $2$-dimensional Lebesgue measure. Restricting to a rotation generated by $q$-th root of unity $ζ$ with all parameters in $\mathbb{Q}(ζ,β)$, it gives a sofic system when $\cos(2π/q) \in \mathbb{Q}(β)$ and $β$ is a Pisot number. It is also shown that the condition $\cos(2π/q) \in \mathbb{Q}(β)$ is necessary by giving a family of non-sofic systems for $q=5$.
Source: arXiv (Cornell University)
Authors: Akiyama, Shigeki, Caalim, Jonathan
Institutions: University of Tsukuba, University of the Philippines Diliman
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W1548243771