It is well known that the set of hypercyclic vectors of a continuous operator acting on a seperable F-space is either empty or a dense \(G_\delta\). Therefore, it is interesting to see how small can the complement of the hypercyclic vectors be. One way to study this problem is through the geometric notion of \(\sigma\)-porous sets.
Inspired by the paper [F. Bayart, Proc. Am. Math. Soc. 133, No. 11, 3309–3316 (2005; Zbl 1202.47009)], the authors studies non-\(\sigma\)-porous subsets of Lebesgue spaces of measurable functions on a locally compact group. As an application, they show the existence of sequences of weighted tanslation operators for which the complement of the set of hypercyclic vectors is non-\(\sigma\)-porous. As a corollary, the authors show the following result for sequences of weighted backward shift operators on Banach sequence spaces:
Corollary. Let \(p \geq 1,\left(\gamma_n\right)_n \subseteq \mathbb{N}\) be strictly increasing and \(\left(w_n\right)_{n \in \mathbb{Z}}\) be a bounded sequence in \((0, \infty)\) such that \[ \left(\frac{1}{w_{\gamma_0} w_{\gamma_1} w_{\gamma_2} \cdots w_{\gamma_n}}\right)_n \in \ell^p(\mathbb{Z}) \] Then, the set of all non-hypercyclic vectors of the sequence \(\left(T_n\right)_n\) is not \(\sigma\)-porous, where \[ \left(T_{n+1} a\right)_k:=w_{\gamma_0} w_{\gamma_1} w_{\gamma_2} \cdots w_{\gamma_n} a_{k+\gamma_{n+1}}, \quad k \in \mathbb{N}_0 \] for all \(a:=\left(a_j\right)_j \in \ell^p(\mathbb{Z})\).