We introduce a numeration system, called the <em>beta Cantor series expansion</em>, that generalizes the classical positive and negative beta expansions by allowing non-integer bases in the Q-Cantor series expansion. In particular, we show that for a fix $\gamma \in \mathbb{R}$ and a sequence $B=\{\beta_i\}$ of real number bases, every element of the interval $x \in [\gamma,\gamma+1)$ has a <em>beta Cantor series expansion</em> with respect to B where the digits are integers in some alphabet $\mathcal{A}(B)$. We give a criterion in determining whether an integer sequence is admissible when $B$ satisfies some condition. We provide a description of the reference strings, namely the expansion of $\gamma$ and $\gamma+1$, used in the admissibility criterion.
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 Q(ζ, β), the map gives rise to a sofic system when cos(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.
We study invariant measures of a piecewise expanding map in R m defined by an expanding similitude modulo lattice. Using the result of Bang [5] on the plank problem of Tarski, we show that when the similarity ratio is not less than m + 1, it has an absolutely continuous invariant measure equivalent to the mdimensional Lebesgue measure, under some mild assumption on the fundamental domain. Applying the method to the case m = 2, we obtain an alternative proof of the result in [1] together with some improvement.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.