A natural extension for the greedy \(\beta\)-transformation with three arbitrary digits
For a real number \(\beta>1\) and finite set of real numbers \(A\) (the digits), an expression of the form \(x=\sum_{n=1}^{\infty}a_n\beta^{-n}\) with \(a_n\in A\) for all \(n\geq1\) is called a \(\beta\)-expansion for \(x\). \textit{M. Pedicini} [Theor. Comput. Sci. 332, No. 1-3, 313--336 (2005; Zbl 1080.11009)] gave conditions on \(A\) to give a range of values of \(x\) that are guaranteed to have a \(\beta\)-expansion, and an algorithm to generate the so-called greedy expansion. Here a planar version of the natural invertible extension of the piecewise-linear map that generates this greedy expansion in the case \(| A|=3\). Using this, a closed formula is found for the density of the unique invariant measure absolutely continuous with respect to Lebesgue measure, and this measure is shown to be exact and weakly Bernoulli.
scientific article; zbMATH DE number 5770413
scientific article