By a *dynamical system* I understand a pair $(K,G)$ consisting a compact Hausdorff space and a subgroup $G$ of the homeomorphism group of $K$.

We say that a dynamical system $(K,G)$

$\bullet$ is *topologically transitive* if for every non-empty open set $U\subseteq K$ its orbit $GU=\{g(x):g\in G,\;x\in U\}$ is dense in $K$;

$\bullet$ *has dense orbit* if for some point $x\in K$ its orbit $Gx$ is dense in $K$.

It is easy to see that a dynamical system is topologically transitive if it has a dense orbit. If the space $K$ is metrizable and nonempty, then the converse also is true.

On the other hand, under $\mathrm{non}(\mathcal M)<\mathfrak c$, there exists a subgroup $G\subset S_\omega$ of cardinality $|G|\le\mathrm{non}(\mathcal M)<\mathfrak c$ that induces a topologically transitive action of the Stone-Cech remainder $\omega^*=\beta\omega\setminus\omega$. The dynamical system $(\omega^*,G)$ does not have dense orbits since the space $\omega^*$ has density $\mathfrak c>|G|$. I am interested if such an example can be constructed in ZFC.

Problem.Is there topologically transitive dynamical system without dense orbits?