Generalized Gauss transformations
For a fixed real number \(p\geq 1,\) let \(T_{p}\) be the transformation on the unit interval defined by \(T_{p}(x)=\{ 1/x^{p}\}\), where \(\{ x\}\) is the fractional part of \(x.\) Let \(\rho_{p}(x)dx\) be the \(T_{p}\)-invariant absolutely continuous ergodic measure. Under these conditions the main result claims that \(\rho_{p}(x)\) converges to \(1\) in the supremum norm as \(p\rightarrow\infty.\) The convergence problem for \(\rho_{p}(x)\) has originally been suggested by \textit{G. H. Choe} [Appl. Math. Comput. 109, 287-299 (2000)], and the author has proved the convergence. Let \([a_{1}, a_{2}, \ldots ]_{T_{p}}\) be a symbolic representation of \(x\in [0,1)\) induced by \(T_{p}.\) It is also shown that the distribution of relative frequency of \(k\in\mathbb{N}\) in \([a_{1}, a_{2}, \ldots]_{T_{p}}\) satisfies the central limit theorem.
scientific article; zbMATH DE number 1957141