Let $\{T_i\}_{i \in I}$ be a family of hypercylic operators on a separable Banach space $X$. From the transitivity theorem, we know that $HC(T_i)$, the set of vectors $x \in X$ with $\{T_i^n(x):n \in \mathbb{N}\}$ dense in $X$, is itself a dense $G_{\delta}$ subset of $X$.
It is known, or when is it the case that:
- There exists a finite-dimensional subspace $Y$ of $X$ such that $$ X-\bigcup_{i \in I} HC(T_i)\subset Y$$
- There is no finite-dimensional subspace $Z\subseteq X$ such that $$ X-\bigcup_{j=1}^n \, HC(T_{i_j})\subseteq Z $$ for any finite sequence $i_1,\dots,i_n$ in $I$
