  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 $\beta$.

Dynamical Systems