Dynamics of one-dimensional maps: Symbols, uniqueness, and dimension

This dissertation is a study of the dynamics of one-dimensional unimodal maps and is mainly concerned with those maps which are trapezoidal. The trapezoidal function, f{dollar}sb{lcub}rm e{rcub}{dollar}, is defined for e {dollar}in{dollar} (0,1/2) by f{dollar}sb{lcub}rm e{rcub}{dollar}(x) = x/e for x {dollar}in{dollar} (0,e), f{dollar}sb{lcub}rm e{rcub}{dollar}(x) = 1 for x {dollar}in{dollar} (e,1 {dollar}-{dollar} e), and f{dollar}sb{lcub}rm e{rcub}{dollar}(x) = (1 {dollar}-{dollar} x)/e for x {dollar}in{dollar} (1 {dollar}-{dollar} e,1). We study the symbolic dynamics of the kneading sequences and relate them to the analytic dynamics of these maps.; Chapter one is an overview of the present theory of Metropolis, Stein, and Stein (MSS). In Chapter two a formula is given that counts the number of MSS sequences of length n. Next, the number of distinct primitive colorings of n beads with two colors, as counted by Gilbert and Riordan, is shown to equal the number of MSS sequences of length n. An algorithm is given that produces a bijection between these two quantities for each n. Lastly, the number of negative orbits of size n for the function f(z) = z{dollar}sp2 -{dollar} 2, as counted by P. J. Myrberg, is shown to equal the number of MSS sequences of length n.; For an MSS sequence P, let H{dollar}sbinfty{dollar}(P) be the unique common extension of the harmonics of P. In Chapter three it is proved that there is exactly one {dollar}lambda{dollar}(P) {dollar}in{dollar} (0,1) such that the itinerary of {dollar}lambda{dollar}(P) under the map {dollar}lambda{dollar}(P)f{dollar}sb{lcub}rm e{rcub}{dollar} is H{dollar}sbinfty{dollar}(P).; In Chapter four it is shown that only period doubling or period halving bifurcations can occur for the family {dollar}lambda{dollar}f{dollar}sb{lcub}rm e{rcub}{dollar}, {dollar}lambdain{dollar} (0,1). Results concerning how the size of a stable orbit changes as bifurcations of the family {dollar}lambda{dollar}f{dollar}sb{lcub}rm e{rcub}{dollar} occur are given.; Let {dollar}lambdain{dollar} (0,1) be such that 1/2 is a periodic point of {dollar}lambda{dollar}f{dollar}sb{lcub}rm e{rcub}{dollar}. In this case 1/2 is superstable. Chapter five investigates the boundary of the basin of attraction of this stable orbit. An algorithm is given that yields a graph directed construction such that the object constructed is the basin boundary. From this we analyze the Hausdorff dimension and measure in that dimension of the boundary. The dimension is related to the simple {dollar}beta{dollar}-numbers, as defined by Parry.