Dynamics of One-Dimensional Maps: Symbols, Uniqueness, and Dimension

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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_e, is defined for eΣ(0,1/2) by f_e(x)=x/e for xΣ[0,e], f_e(x)=1 for xΣ(e,1-e), and f_e(x)=(1-x)/e for xΣ[1-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 ... continued below

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v, 106 leaves : ill.

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Brucks, Karen M. (Karen Marie), 1957- May 1988.

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  • Brucks, Karen M. (Karen Marie), 1957-

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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_e, is defined for eΣ(0,1/2) by f_e(x)=x/e for xΣ[0,e], f_e(x)=1 for xΣ(e,1-e), and f_e(x)=(1-x)/e for xΣ[1-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 bisection between these two quantities for each n. Lastly, the number of negative orbits of size n for the function f(z)=z^2-2, 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_ϖ(P) be the unique common extension of the harmonics of P. In Chapter three it is proved that there is exactly one J(P)Σ[0,1] such that the itinerary of λ(P) under the map is λ(P)f_e is H_ϖ(P).
In Chapter four it is shown that only period doubling or period halving bifurcations can occur for the family λf_e, λΣ[0,1]. Results concerning how the size of a stable orbit changes as bifurcations of the family λf_e occur are given.
Let λΣ[0,1] be such that 1/2 is a periodic point of λf_e. 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 β-numbers, as defined by Parry.

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v, 106 leaves : ill.

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  • May 1988

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  • Aug. 22, 2014, 6 p.m.

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  • Dec. 1, 2015, 10:36 a.m.

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Citations, Rights, Re-Use

Brucks, Karen M. (Karen Marie), 1957-. Dynamics of One-Dimensional Maps: Symbols, Uniqueness, and Dimension, dissertation, May 1988; Denton, Texas. (digital.library.unt.edu/ark:/67531/metadc332102/: accessed August 24, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; .