Let $X$ be a separable Hausdorff topological space and $\phi \in C(X,X)$ be a topologically transitive map. Further, let $V$ be a fixed non-empty open subset of $X$. Then does there necessarily exist a countable open cover $\{U_i\}_{i \in \mathbb{N}}$ of $X$ and a sequence of natural numbers $\{N_i\}_{i \in \mathbb{N}}$ such that $$ \bigcup_{n \in \mathbb{N}} \phi^{N_i}(U_i) \subseteq V? $$

ie the entire space fits into $V$; eventually?

[$\phi$ is said to be *topologically transitive* iff for every two non-empty open subsets $U,V\subseteq X$ there exists some $N\in \mathbb{N}$ such that
$\phi^N(U)\cap V \neq \emptyset$.]