Let $G$ be a subgroup of the permutation group $S\_\omega$ of the countable infinite set $\omega$. Each bijection $g\in G$ admits a unique extension to a homeomorphism $\bar g$ of the Stone-Cech compactification $\beta\omega$ of $\omega$. The homeomorphism $\bar g$ induces a homeomorphism of the remainder $\omega^\*=\beta\omega\setminus\omega$ of the Stone-Cech compactification. So, we obtain a continuous action of the group $G$ on the compact Hausdorff space $\omega^\*$. I am interested in properties of the obtained dynamical system $(\omega^\*,G)$. Namely, I would like to know the answer to the following



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> **Problem.** Is there a subgroup $G\subseteq S\_\omega$ such that the dynamical system $(\omega^\*,G)$ is topologically transitive (=each nonempty open set has dense orbit) but does not have a dense orbit.
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An example of such subgroup $G$ exists under the assumption $\mathrm{non}(\mathcal M)<\mathfrak c$. So, the question actually ask about the situation in ZFC.


**Remark.** If a group $G\subseteq S\_\omega$ induces a topologically transitive action on $\omega^\*$, then $G$ has large cardinality, namely, $|G|\ge\mathsf \Sigma\ge\max\{\mathfrak b,\mathfrak s,\mathrm{cov}(\mathcal M)\}$. More information on the cardinal $\mathsf \Sigma$ can be found in [this preprint](https://arxiv.org/pdf/2002.08800).



A permutation group inducing a topologically transitive action without dense orbits on $\omega^*$
It turns out that this problem is independent of ZFC because of the following simple



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> **Theorem.** Under $\mathfrak t=\mathfrak c$, every topologically transitive continuous action of a group $G$ on $\omega^\*$ has a dense orbit.
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*Proof.* Let $(A\_\alpha)\_{\alpha\in\mathfrak c}$ be an enumeration of all infinite subsets of $\omega$. By transfinite induction we shall construct a transfinite sequence of infinite subsets $(U\_\alpha)\_{\alpha\in\mathfrak c}$ of $\omega$ and a transfinite sequence $(g\_\alpha)\_{\alpha\in\mathfrak c}$ of elements of the group $G$ such that for every $\alpha\in\mathfrak c$ the following conditions are satisfied:


(a) $U\_\alpha\subseteq^\* U\_\beta$ for all $\beta<\alpha$;


(b) $g\_\alpha(U\_\alpha)\subseteq^\* A\_\alpha$.


To start the inductive construction, put $U\_0=A\_0$ and $g\_0$ be the identity of the group $G$. Assume that for some ordinal $\alpha\in\mathfrak c$, a transfinite sequence $(U\_\beta)\_{\beta<\alpha}$ satisfying the condition (a) has been constructed. By the definition of the [tower number](http://www.math.lsa.umich.edu/~ablass/hbk.pdf) $\mathfrak t$ and the equality $\mathfrak t=\mathfrak c>\alpha$, there exists an infinite subset $V\_\alpha\subseteq\omega$ such that $V\_\alpha\subseteq^\* U\_\beta$ for all $\beta<\alpha$. The infinite sets $V\_\alpha$ and $A\_\alpha$ and determine clopen sets $\overline V\_\alpha=\{p\in\omega^\*:V\_\alpha\in p\}$ and $\bar A\_\alpha=\{p\in\omega^\*:A\_\alpha\in p\}$ in the space $\omega^\*=\beta\omega\setminus\omega$. Since the action of the group $G$ on $\omega^\*$ is topologically transitive, there exist $g\_\alpha$ and an infinite subset $U\_\alpha\subset V\_\alpha$ such that $g\_\alpha(\overline U\_\alpha)\subseteq \bar A\_\alpha$, which implies $g\_\alpha (U\_\alpha)\subseteq^\* A\_\alpha$. This completes the inductive step.


Adter completing the inductive construction, extend the family $\{U\_\alpha\}\_{\alpha\in\mathfrak c}$ to a free ultrafilter $\mathcal U$ and observe that its orbit intersetcs each clopen set $\bar A\_\alpha$, $\alpha\in\mathfrak c$ and hence is dense in $\omega^\*$. $\qquad\square$


