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](https://www.researchgate.net/publication/251140179_Existence_of_a_dense_orbit_and_topological_transitivity_When_are_they_equivalent). 


On the other hand, under $\mathrm{non}(\mathcal M)<\mathfrak c$, [there exists](https://arxiv.org/pdf/2002.08800) 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?
> 
> 
> 



A topologically transitive dynamical system without dense orbits
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:\ $


$$ \forall\_{n\in\Bbb Z}\forall\_{x\in\Bbb Z}\quad
 s\_n(x):= x+n $$


Let $\ p:=(p\_n)\in K\ $ be arbitrary, and let


$$ P:=\{p\_n:n\in \Bbb Z\} $$


Then there exists non-empty open $\ U\ $ in $\ X,\ $ disjoint with set $\ P.\ $ Then non-empty open in $\ K\ $
set $\ W,$


$$ W\ :=\ \pi\_0^{-1}(U)\ $$


is disjoint with the orbit $\ p.\ $


On the other hand, let $\ \emptyset\ne G\subseteq K, $
where $\ G\ $ is open in $\ K.\ $ Then there exists
non-empty $\ H\ $ and integer $ a\ge 0\ $ such that
$\ H\ $ is an open subset of $\ X^{(-a)..a}\ $ (*Perl notation* "s..t") and


$$ \emptyset\ \ne\ \pi\_{(-a)..a}^{-1}(H)\ \subseteq G $$


Obviously, the orbit of $\ \pi\_{(-a)..a}^{-1}(H)\ $, hence of $\ G,\ $ is dense in $\ K.$


***Great!***


