(1) For $\beta>1$ let $T_{\beta}:[0,1)\to [0,1)$ be given by $T_{\beta}(x)=\{\beta x\}$, where $\{\cdot\}$ denotes the non-integer part of real number. It is classical result (due to Parry I think) that the dynamical system $([0,1),T_{\beta})$ is mixing with topological entropy $\log(\beta)$. Now I wonder if this System is Devaney chaotic, means that in addition to transitivity periodic orbits are dense. If $\beta$ is an integer this is well known. In the literature I found the result that the system $([0,1),T_{\beta})$ is sofic if $\beta$ is a Pisot number, an algebraic integer with all its conjugates insides the unite circle. From this we get a dense set of periodic orbits. Is this known for any other $\beta>1$?

(2) For $\mu\in (0,4]$ let $f_{\mu}:[0,1]\to[0,1]$ be given by $f_{\mu}(x)=\mu x(1-x)$. I know the celebrated result that there is a set $B$ of positiv Lebesgue measure such that $([0,1],f_{\mu})$ has an chaotic attractor for $\mu\in B$. If I am not mistaken chaos means here topological and not Devaney chaos. My question is for which algebraic numbers $\mu<4$ the system $([0,1],f_{\mu})$ has a Devaney chaotic subsystem? Perhaps this is a difficult question, is anything known?\

Question (1) is answered. Concerning (2) I understand now that we have Devaney chaos of $([0,1],f_{\mu})$ for all $\mu\in B$. Are there any explicitly known (algebraic) numbers in $B$?