Generalized Gauss transformations
type: Item
instance of: scholarly article
title: Generalized Gauss transformations
zbMATH Open document ID: 1026.60023
DOI: 10.1016/S0096-3003(02)00287-4
author: Young-Ho Ahn
published in: Applied Mathematics and Computation
publication date: 2003-07-30T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 60F05
zbMATH DE Number: 1957141
zbMATH Keywords: Frobenius-Perron operator
zbMATH Keywords: coboundary
zbMATH Keywords: invariant measure
zbMATH Keywords: exact transformation
zbMATH Keywords: Gauss transformation
zbMATH Keywords: eventually expansive
reviewed by: Tae Il Jeon
MaRDI profile type: Publication
cites work: Q4357052
cites work: Generalized continued fractions.
cites work: Théorie ergodique pour des classes d'opérations non completement continues
cites work: Mixing for Markov operators
cites work: Q4037014
cites work: Q4744504
cites work: Bounded variation and invariant measures
full work available at URL: s0096-3003(02)00287-4
OpenAlex ID: W1970371703
description: scientific article; zbMATH DE number 1957141