A point is normal for almost all maps βx plus α mod 1 or generalized β-transformations

We consider the map T(alpha,beta)(x) := beta x + alpha mod 1, which admits a unique probability measure of maximal entropy. For x is an element of [0, 1], we show that the orbit of x is mu(alpha,beta)-normal for almost all (alpha, beta) is an element of [0, 1) x ( 1, infinity) (with respect to Lebesgue measure). Nevertheless, we construct analytic curves in [0, 1) x (1, infinity) along which the orbit of x = 0 is mu(alpha,beta)-normal at no more than one point. These curves are disjoint and fill the set [0, 1) x (1, infinity). We also study the generalized-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 beta.