# Topologically transitive dynamical system mapping space into ball

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 $$\phi^N(U)\cap V \neq \emptyset$$.]

Let $$\phi$$ be the left shift on the set $$X = \{0,1\}^\mathbb{Z}$$ of bi-infinite binary sequences with the prodiscrete topology, and let $$V = \{ x \in X : x_0 = 0 \}$$ be the set of sequences that have $$0$$ at the central coordinate. Then $$X$$ is Hausdorff (even metrizable and compact), $$\phi$$ is transitive (even mixing) and $$V$$ is nonempty and open (even clopen). The all-$$1$$ sequence $$x = \ldots 1 1 1 \ldots$$ satisfies $$\phi^n(x) \notin V$$ for all $$n \in \mathbb{Z}$$. Hence in any open cover $$(U_i)_{i \in I}$$, some $$U_i$$ contains $$x$$ and then $$\phi^n(U_i)$$ is not a subset of $$V$$ for any $$n$$.