### The Classification of Critically Preperiodic Polynomials as Dynamical Systems

Ben Bielefeld, Yuval Fisher, John Hubbard
1992 Journal of The American Mathematical Society
Symbolic dynamics of unimodal mappings: kneading sequences. The combinatorial approach to the dynamics of mappings, which this paper develops, starts with kneading sequences, as developed in [MT] and [CE]. Choose a < c < b E R., and set I = [a, b] and consider first unimodal maps, which we will take to * * This theorem can be used to compute the symbolic sequence from the external angle, or the external angle from the symbolic sequence once Go is chosen; the two choices correspond to the ray in
more » ... spond to the ray in the upper half-plane if Go = 0 and its complex conjugate in the lower half-plane if Go = 1 . Example. This fact can be used to compute the arccosine inductively. For the polynomial p(z) = z2 -2, we saw that ¢p(z) = z + 1/ z, so that ¢p(i 1liO ) = 2cos(2nO). If the symbolic sequence of x = 2cos(2nO) is Sp(x) = (so' sl' ... ), and 0 is written in base 2 according to the table above with first digit 0, then the branch of arccos(x/2) in [0, n] is given by arccos(x/2) = 2nO. Choosing