||Rotational beta expansion: ergodicity and soficness
AKIYAMA, ShigekiCAALIM, Jonathan
Journal of the ｍathematical society of japan
415 , 2017-01 , The Mathematical Society of Japan
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 B1 and B2 depending only on the fundamental domain that if β > B1 then the expanding map has a unique absolutely continuous invariant probability measure, and if β > B2 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 Q(ζ,β), the map gives rise to a sofic system when os(2π/q)∈Q(β) and β is a Pisot number. It is also shown that the condition cos(2π/q)∈Q(β) is necessary by giving a family of non-sofic systems for q=5.