Maps with dense orbits: Ansari's theorem revisited and the infinite torus
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 \(\mathbb{T}^\infty\).
scientific article; zbMATH DE number 5612293