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 mean continuous mappings f:(2) f is monotone decreasing (or increasing) on [a, c];(3) f is monotone increasing (or decreasing, respectively) on [c, b] .We apologize to readers used to unimodal maps with maxima; monic polynomials are best adapted to our purposes. Thus typical unimodal mappings are elements of the quadratic family pc(x) = X2 + c with the intervaland c = O.In this case, the kneading sequence of a point x E

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 mean continuous mappings f: I-I satisfying (1) f(a) = f(b) = b; (2) f is monotone decreasing (or increasing) on [a, c]; (3) f is monotone increasing (or decreasing, respectively) on [c, b]. We apologize to readers used to unimodal maps with maxima; monic polynomials are best adapted to our purposes. Thus typical unimodal mappings are elements of the quadratic family pc(x) = X2 + c with the interval 1= [-(1 + y'1-4c)j2, (1 + y'1-4c)j2] and c = O. In this case, the kneading sequence of a point x E I is the sequence Sj(x) = (so' Sl ' •••) with Sj E {P, R, C}, determined by whether /(x) is on the orientation-preserving (i.e., increasing) side of the "critical point" c, the orientation-reversing (Le., decreasing) side of the "critical point," or on the "critical point," respectively. The central questions are then: 1.1. When is x determined by the sequence Sj(x)?; 1.2. For what families of unimodal maps is f determined by S j(c)? More specifically, is a quadratic polynomial determined by the symbolic sequence of the critical point? These questions and their relatives are the key questions of this paper, and in fact of the whole subject; we answer them completely for preperiodic polynomials. Example. Consider the graphs of unimodal maps in Figure 1.3, showing the forward orbit of the critical point. Both candidate mappings are unimodal FIGURE 1.3. Two unimodal mappings with x j + 1 = f(x j).

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