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 $beta$.