Journal Article Rotational beta expansion: ergodicity and soficness

AKIYAMA, Shigeki  ,  CAALIM, Jonathan

69 ( 1 )  , pp.397 - 415 , 2017-01 , The Mathematical Society of Japan
ISSN:0025-5645
NCID:AA0070177X
Description
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.
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