A Point is Normal for Almost All Maps $\beta x + \alpha \mod 1$ or Generalized $\beta$-Maps

We consider the map $T_{\alpha,\beta}(x):= \beta x + \alpha \mod 1$, which admits a unique probability measure of maximal entropy $\mu_{\alpha,\beta}$. For $x \in [0,1]$, we show that the orbit of $x$ is $\mu_{\alpha,\beta}$-normal for almost all $(\alpha,\beta)\in[0,1)\times(1,\infty)$ (Lebesgue measure). Nevertheless we construct analytic curves in $[0,1)\times(1,\infty)$ along them the orbit of $x=0$ is at most at one point $\mu_{\alpha,\beta}$-normal. These curves are disjoint and they fill the set $[0,1)\times(1,\infty)$. We also study the generalized $\beta$-maps (in particular the tent map). We show that the critical orbit $x=1$ is normal with respect to the measure of maximal entropy for almost all $\beta$.