Subsemigroups of transitive semigroups (Q2908170)

The present paper is concerned with the following problem in topological dynamics, which has its motivations in linear dynamics, i.e., the dynamics of continuous linear operators acting on a topological vector space. Let \(X\) be a topological space, and let \(\Gamma \) be a topological semigroup acting continuously on \(X\). Let \(\Gamma _{0}\) be a subsemigroup of \(\Gamma \). Assuming that \(\Gamma \) has a transitive point, \(x\in X\) (i.e., that the set \(\Gamma .x\) is dense in \(X\)), when is it possible to deduce that \(x\) is also a transitive point for the subsemigroup \(\Gamma _{0}\)? Two general results (Theorem 1.1 and Theorem 1.2) are proved concerning this question, which allow the author to retrieve all known results in linear dynamics which fit in this setting, the two main ones being:NEWLINENEWLINE(1) the Conejero-Müller-Peris theorem of [\textit{J. A. Conejero} et al., J. Funct. Anal. 244, No. 1, 342--348 (2007; Zbl 1123.47010)] which states that every operator in a hypercyclic one parameter semigroup \((T_{t})_{t\geq 0}\) is hypercyclic;NEWLINENEWLINE(2) the following general result of \textit{S. Shkarin} [J. Math. Anal. Appl. 348, No. 1, 193--210 (2008; Zbl 1148.47009)]: if \(T\) is a hypercyclic operator acting on a topological vector space \(X\) and if \(g\) is a topological generator of a compact group \(G\), then if \(x\) is any hypercyclic vector for \(T\), the set \(\{(g^{n},T^{n}x):n\in \mathbb{N}\}\) is dense in \(G\times X\).