We say that dynamical $(X,f)$ system is transitive, if there is $x\in X$, such that $x$ has a dense orbit.
I was looking for the example of the dynamical system $(X,\varphi)$, such that it's transitive, but $(X,\varphi^{2})$ is not. In some articles I found an example:
$X=[0,1]$, $\ \ \ \ \varphi(x)=\begin{cases} \frac{1}{2}+2x \ \ for \ \ 0\le x\le \frac{1}{4} \\ \frac{3}{2}-2x \ \ for \ \ \frac{1}{4}\le x\le\frac{1}{2} \\ 1-x \ \ for \ \ \frac{1}{2}\le x\le 1 \end{cases}$
I see that, since intervals $[0,\frac{1}{2}]$ and $[\frac{1}{2},1]$ are $\varphi^{2}$-invariant, so we don't have any point with dense orbit.
But the fact, that $(X,\varphi)$ is transitive, is just mentioned, like some obvious thing, without any explanations. I tried to check, what points are able to have a dense orbit, but without any results. I was thinking about irrational numebrs, I only showed that they are not periodic, but it isn't enough.
If anyone would have some ideas, I'd be greatful.