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The answer is yes: there is a topologically transitive dynamical system without dense orbits. Indeed, let X be a topological space that is not separable. Let $\ K=X^{\Bbb Z},\$ and let $\ G\$ be the group of homeomorphism of $\ K,\$ induced by shifts $\ s_n\ (n\in\Bbb Z)\$ of $\ \Bbb Z:\$  \forall_{n\in\Bbb Z}\forall_{x\in\Bbb Z}\quad ...
It turns out that this problem is independent of ZFC because of the following simple Theorem. Under $\mathfrak t=\mathfrak c$, every topologically transitive continuous action of a group $G$ on $\omega^*$ has a dense orbit. Proof. Let $(A_\alpha)_{\alpha\in\mathfrak c}$ be an enumeration of all infinite subsets of $\omega$. By transfinite induction we ...