Let be $T_{\beta}:[0,1]\to [0,1]$ defined by $T_{\beta}(x)=\beta x \bmod 1$ where $\beta \in (1,2).$
Questions:
$T_{\beta}$ is topologically transitive?
What about the periodic points?
$T_{\beta}$ is topologically mixing ?
The answers to this question when beta is equal to 2 comes from the fact that $ T_{\beta}$ is conjugated to two side shift.
However in this case $ \beta \in (1,2).$ So my attempt is in brute force, i.e, I ventured to say that the points $x\in \mathbb{R}\setminus \mathbb{Q}$ has dense orbit, since the orbit of $x $ is the set
$\operatorname{Orb}(x)=\{T^n_{\beta}(x), ~~n\in\mathbb{N}\}\;,$ and
$T^n(x)=\beta^n(x)\bmod 1$
that is, $T^n(x)=\beta^nx+l,~~~l\in\mathbb{Z},~~n\in \mathbb{N}.$ think the above set is dense, for sets of the form $A=\{nx+m,~~ ~~~l\in\mathbb{Z},~~n\in \mathbb{N}\}$ when $x$ is irrational. this is not quite a proof ... I'm only showing where my intuition is guiding me.
I wonder if anyone can do it more elegantly.