Ergodicity of \(N\)-continued fraction expansions
The authors of this paper study a special kind of continued fraction expansions, firstly introduced by \textit{E. B. Burger} and his co-authors in the paper [J. Number Theory 128, No. 1, 144--153 (2008; Zbl 1210.11078)]. Applying the map \(T_{N}:[0,1)\rightarrow [0,1)\), defined by \(T_{N}(x)=\frac{N}{x}-\lfloor\frac{N}{x}\rfloor\) with \(x\neq0\) and \(T_{N}(0)=0\), every \(x\in \mathbb{R}\) can be written in the form \[ x=n_{0}+\frac{N}{n_{1}+\frac{N}{n_{2}+\ddots +\frac{N}{n_{k}+\ddots}}}=[n_{0};n_{1},n_{2},\ldots,n_{k}\ldots]_{N}, \] where \(N\in\mathbb{R}\) and \(N \neq 0\). The first theorem (Theorem 3.1) is: Let \(N\in \mathbb{N}\). Then the dynamical system \(([0,1),\mathcal{B},\mu_{N},T_{N})\) is exact and therefore ergodic. Here \(\mathcal{B}\) is the Borel \(\sigma\)-algebra on \([0,1)\), and \(\mu_{N}\) is a \(T_{N}\)-invariant probability measure on \(([0,1), \mathcal{B})\) with density \(\frac{1}{\log\frac{N+1}{N}}\frac{1}{N+x}1_{[0,1)}(x)\). Other theorem with the same conditions (Theorem 3.3) asserts that the dynamical system \(([0,1),\mathcal{B},\mu_{-N},T_{-N})\) is also ergodic. In particular case \(N \geq 2\), \(\mu_{-N}\) is a probability measure with density \(\frac{1}{\log\frac{N}{N-1}}\frac{1}{N-x}1_{[0,1)}(x)\). Similar theorems with other conditions are proved and many examples are discussed. Special attention is turned to the periodicity of digits and conjectures on the base of calculation and their histograms.
scientific article; zbMATH DE number 6341801
scientific article