A Point is Normal for Almost All Maps beta x+ alpha mod 1 or Generalised beta-Transformations
We consider the map T-alpha,T-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.