We consider the map Tα,β(x):=βx+αmod1, which admits a unique probability measure μα,β of maximal entropy. For x[0,1], we show that the orbit of x is μα,β-normal for almost all (α,β)[0,1)×(1,∞) (with respect to Lebesgue measure). Nevertheless, we construct analytic curves in [0,1)×(1,∞) along which the orbit of x=0 is μα,β-normal at no more than one point. These curves are disjoint and fill the set [0,1)×(1,∞). We also study the generalized β-transformations (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
