Algebraic Irrational Binary Numbers Cannot Be Fixed Points of Non-trivial Constant Length or Primitive Morphisms

How random are the digits of an algebraic irrational number in a given base? A common conjectured answer to this vague question is that these digits are ``really random.'' For example, define a normal number in base k 2 to be a real number x such that, for each integer d 1, each block of length d occurs in the k-ary expansion of x with frequency 1 k. A widely believed conjecture is that an algebraic irrational number is a normal number in each base k 2. This conjecture is open and seems really out of reach. An alternative formulation of this idea is that the digits occurring in the k-ary expansion of an algebraic irrational number cannot be obtained via a simple algorithm. For example the Champernowne number is obtained by concatenating the decimal expansions of the consecutive integers, i.e.,