Maps with dense orbits: Ansari's theorem revisited and the infinite torus
type: Item
instance of: scholarly article
title: Maps with dense orbits: Ansari's theorem revisited and the infinite torus
zbMATH Open document ID: 1187.47011
author: Miguel Marano
author: Héctor N. Salas
published in: Bulletin of the Belgian Mathematical Society - Simon Stevin
publication date: 2009-10-09T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.bbms/1251832374
review text: This paper consists of two parts. In the first one, the authors present a natural proof of a theorem of \textit{S.\,I.\thinspace Ansari} [J.~Funct.\ Anal.\ 128, No.\,2, 374--383 (1995; Zbl 0853.47013)] about the iterates \(T^n\) of a map \(T\) with dense orbit. The proof presented here is inspired by a result of Bourdon for the case of \(T^2\). An example is given to show that for each prime number \(p\) there is a connected metric space \(X\) without isolated points and a map \(f:X \to X\) with a dense orbit such that \(f^p\) does not have a dense orbit. The second part is motivated by a theorem due to \textit{F.\,León-Saavedra} and \textit{V.\,Müller} [Integral Equations Oper.\ Theory 50, No.\,3, 385--391 (2004; Zbl 1079.47013)], stating that if \(T\) is a bounded linear operator on a complex Banach space with a dense orbit, then \(\lambda T\) has also a dense orbit for all complex numbers \(\lambda\) such that \(|\lambda|=1\). Nonlinear analogues of this result are investigated for endomorphisms on the infinite torus \(\mathbb{T}^\infty\).
reviewed by: José Bonet
Mathematics Subject Classification ID: 47A16
Mathematics Subject Classification ID: 37A99
Mathematics Subject Classification ID: 22D40
zbMATH DE Number: 5612293
zbMATH Keywords: hypercyclic operators
zbMATH Keywords: dense orbits
zbMATH Keywords: transitive maps
zbMATH Keywords: measurable preserving maps
zbMATH Keywords: ergodic maps
zbMATH Keywords: infinite torus
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 5612293